Monitoring Progress And Modeling With Mathematics 3.2 Answers

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Big Ideas Math Book Algebra 1 Answer Key Chapter 3 Graphing Linear Functions

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• Graphing Linear Functions Maintaining Mathematical Proficiency – Page 101
• Graphing Linear Functions Mathematical Practices – Page 102
• Lesson 3.1 Functions – Page (103 – 110)
• Functions 3.1 Exercises – Page(108-110)
• Lesson 3.2 Linear Functions – Page(111-120)
• Linear Functions 3.2 Exercises – Page(117-120)
• Lesson 3.3 Function Notation – Page(121-126)
• Function Notation 3.3 Exercises – Page(125-126)
• Graphing Linear Functions Study Skills: Staying Focused During Class – Page 127
• Graphing Linear Functions 3.1 – 3.3 Quiz – Page 128
• Lesson 3.4 Graphing Linear Equations in Standard Form – Page(129-134)
• Graphing Linear Equations in Standard Form 3.4 Exercises – Page(133-134)
• Lesson 3.5 Graphing Linear Equations in Slope-Intercept Form – Page(135-144)
• Graphing Linear Equations in Slope-Intercept Form 3.5 Exercises – Page(141-144)
• Lesson 3.6 Transformations of Graphs of Linear Functions – Page(145-154)
• Transformations of Graphs of Linear Functions 3.6 Exercises – Page(151-154)
• Lesson 3.7 Graphing Absolute Value Functions – Page(155-162)
• Graphing Absolute Value Functions 3.7 Exercises – Page(160-162)
• Graphing Linear Functions Performance Task: The Cost of a T-Shirt – Page 163
• Graphing Linear Functions Chapter Review – Page(164-168)
• Graphing Linear Functions Chapter Test – Page 169
• Graphing Linear Functions Cumulative Assessment – Page (170- 171)

Graphing Linear Functions Maintaining Mathematical Proficiency

Plot the point in a coordinate plane. Describe the location of the point.

Question 1.
A(3, 2)
Answer:
The given point is: A (3,2)
Compare the given point with (x, y)
So,
x = 3 and y = 2
Hence,
The representation of the point in the coordinate plane is:

Question 2.
B(-5, 1)
Answer:
The given point is: B (-5,1)
Compare the given point with (x, y)
So,
x = -5 and y = 1
Hence,
The representation of the given point in the coordinate plane is:

Question 3.
C(0, 3)
Answer:
The given point is: c (0,3)
Compare the given point with (x,y)
So,
x = 0 and y = 3
Hence,
The representation of the given point in the coordinate plane is:

Question 4.
D(-1, -4)
Answer:
The given point is: D (-1,-4)
Compare the given point with (x,y)
So,
x = -1 and y = -4
Hence,
The representation of the given point in the coordinate plane is:

Question 5.
E(-3, 0)
Answer:
The given point is: E (-3,0)
Compare the given point with (x,y)
So,
x = -3 and y = 0
Hence,
The representation of the given point in the coordinate plane is:

Question 6.
F(2, -1)
Answer:
The given point is: F (2,-1)
Compare the given point with (x,y)
So,
x = 2 and y = -1
Hence,
The representation of the given point in the coordinate plane is:

Evaluate the expression for the given value of x.

Question 7.
3x – 4; x = 7
Answer:
The value of the expression for the given value of x is: 17

Explanation:
The given expression is:
3x – 4 with x = 7
Hence,
The value of the expression is:
3 (7) – 4 = 21 – 4 = 17
Hence, from the above,
We can conclude that the value of the expression for the given value of x is: 17

Question 8.
-5x + 8; x = 3
Answer:
The value of the expression for the given value of x is: -7

Explanation:
The given expression is:
-5x + 8 with x = 3
Hence,
The value of the expression is:
-5 (3) + 8 = -15 + 8 = -7
Hence, from the above,
We can conclude that the value of the expression for the given value of x is: -7

Question 9.
10x + 18; x = 5
Answer:
The value of the expression for the given value of x is: 68

Explanation:
The given expression is:
10x + 18 with x = 5
Hence,
The value of the expression is:
10 (5) + 18 = 50 + 18 = 68
Hence, from the above,
We can conclude that the value of the expression for the given value of x is: 68

Question 10.
-9x – 2; x = -4
Answer:
The value of the expression for the given value of x is: 34

Explanation:
The given expression is:
-9x – 2 with x = -4
Hence,
The value of the expression is:
-9 (-4) – 2 = 36 – 2 = 34
Hence, from the above,
We can conclude that the value of the expression for the given value of x is: 34

Question 11.
24 – 8x; x = -2
Answer:
The value of the expression for the given value of x is: 40

Explanation:
The given expression is:
24 – 8x with x = -2
Hence,
The value of the expression is:
24 – 8 (-2) = 24 + 16 = 40
Hence, from the above,
We can conclude that the value of the expression for the given value of x is: 40

Question 12.
15x + 9; x = -1
Answer:
The value of the expression for the given value of x is: -6

Explanation:
The given expression is:
15x + 9 with x = -1
Hence,
The value of the expression is:
15 (-1) + 9 = -15 + 9 = -6
Hence, from the above,
We can conclude that the value of the expression for the given value of x is: -6

Question 13.
ABSTRACT REASONING
Let a and b be positive real numbers. Describe how to plot (a, b), (-a, b), (a, -b), and (-a, -b).
Answer:
It is given that a and b are positive real numbers
The given points are (a, b), (-a, b), (a, -b) and (-a, -b)
Let the names of the points be:
A (a, b), B (-a, b), C (a, -b), and D (-a, -b)
We know that,
The coordinate plane is divided into 4 parts. These parts are called  "Quadrants"
So,
The representation of a and b in the 4 quadrants are:
1st Quadrant: (a, b)
2nd Quadrant: (-a, b)
3rd Quadrant: (-a, -b)
4th Quadrant: (a, -b)
Hence,
The representation of the given points in the coordinate plane is:

Graphing Linear Functions Mathematical Practices

Monitoring Progress

Determine whether the viewing window is square. Explain.

Question 1.
-8 ≤ x ≤ 7, -3 ≤ y ≤ 7

Question 2.
-6 ≤ x ≤ 6, -9 ≤ y ≤ 9

Question 3.
-18 ≤ x ≤ 18, -12 ≤ y ≤ 12

Use a graphing calculator to graph the equation. Use a square viewing window.

Question 4.
y = x + 3
Answer:
The given equation is:
y = x + 3
Now,
We can find the values of x and y by putting the values 0, 1, 2…..
Hence,
The representation of the given equation in the coordinate plane is:

Question 5.
y = -x – 2
Answer:
The given equation is:
y = -x – 2
Now,
We can find the values of x and y by putting the values 0, 1, 2…..
Hence,
The representation of the given equation in the coordinate plane is:

Question 6.
y = 2x – 1
Answer:
The given equation is:
y = 2x – 1
Now,
We can find the values of x and y by putting the values 0, 1, 2…..
Hence,
The representation of the given equation in the coordinate plane is:

Question 7.
y = -2x + 1
Answer:
The given equation is:
y = -2x + 1
Now,
We can find the values of x and y by putting the values 0, 1, 2…..
Hence,
The representation of the given equation in the coordinate plane is:

Question 8.
y = –\(\frac{1}{3}\)x – 4
Answer:
The given equation is:
y = –\(\frac{1}{3}\)x – 4
Now,
We can find the values of x and y by putting the values 0, 1, 2…..
Hence,
The representation of the given equation in the coordinate plane is:

Question 9.
y = \(\frac{1}{3}\)x + 2
Answer:
The given equation is:
y = \(\frac{1}{3}\)x + 2
Now,
We can find the values of x and y by putting the values 0, 1, 2…..
Hence,
The representation of the given equation in the coordinate plane is:

Question 10.
How does the appearance of the slope of a line change between a standard viewing window and a square viewing window?
Answer:
A typical graphing calculator screen has a height to width ratio of 2 to 3. This means that when you use the standard viewing window of -10 to 10 ( on each axis ), the graph will not be in its true perspective.
To see a graph in its true perspective, you need to use a square viewing window, in which the tick marks on the x-axis are spaced the same as the tick marks on the y-axis.

Lesson 3.1 Functions

Essential Question

A relation pairs inputs with outputs. When a relation is given as ordered pairs, the x-coordinates are inputs and the y-coordinates are outputs. A relation that pairs each input with exactly one output is a function.

EXPLORATION 1
Describing a Function
Work with a partner. Functions can be described in many ways.

• by an equation
• by an input-output table
• using words
• by a graph
• as a set of ordered pairs

a. Explain why the graph is shown represents a function.
Answer:
The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because that x value has more than one output. A function has only one output value for each input value.

b. Describe the function in two other ways.
Answer:
The function can be described in 4 ways. They are:
a. A function can be represented verbally.
Example:
The circumference of a square is four times one of its sides.
b. A function can be represented algebraically.
Example:
3 x + 6 .
c. A function can be represented numerically.
d. A function can be represented graphically.

EXPLORATION 2

Identifying Functions
Work with a partner. Determine whether each relation represents a function. Explain your reasoning.

e. (-2, 5), (-1, 8), (0, 6), (1, 6), (2, 7)
f. (-2, 0), (-1, 0), (-1, 1), (0, 1), (1, 2), (2, 2)
g. Each radio frequency x in a listening area has exactly one radio station y.
h. The same television station x can be found on more than one channel y.
i. x = 2
j. y = 2x + 3
Answer:
We know that,
Functions can be described in different ways. They are:
a. By an equation
b. By an input-output table
c. Using words
d. By a graph
e. As a set of ordered pairs
We know that,
A function has only a single input for each or multiple outputs
Hence,
The given Exercises above i.e., a, e, g,  j are the relations that represent a function and the remaining relations are not functions

Communicate Your Answer

Question 3.
What is a function? Give examples of relations, other than those in Explorations 1 and 2, that (a) are functions and (b) are not functions.
Answer:
Definition of Function:
A relation from a set of inputs to a set of possible outputs where each input is related to exactly one output is known as " Function"
Examples of relation that is a function:
a. y = x + 3
b. y = -x² + 1
Example of relation that is not a function:

We know that,
A relation is a set of inputs for specific outputs.
But from the above,
A single input has multiple outputs

3.1 Lesson

Monitoring Progress

Determine whether the relation is a function. Explain.

Question 1.
(-5, 0), (0, 0), (5, 0), (5, 10)
Answer:
The given relation is:
(-5, 0), (0, 0), (5, 0), (5, 10)
From the given relation,
We know that,
x represents the input
y represents the output
We know that,
For a function, each input should correspond with only one output
But,
When we observe the given relation, there are 2 outputs for a single input
Hence, from the above,
We can conclude that the given relation is not a function

Question 2.
(-4, 8), (-1, 2), (2, -4), (5, -10)
Answer:
The given relation is:
(-4, 8), (-1, 2), (2, -4), (5, -10)
From the given relation,
We know that,
x represents the input
y represents the output
We know that,
For a function, each input should correspond with only one output
Hence, from the above,
We can conclude that the given relation is a function

Question 3.

Answer:
The given table is:

The representation of the table in the form of relation is:
(2, 2.6), (4, 5.2), (6, 7.8)
From the given relation,
We know that,
x represents the input
y represents the output
We know that,
For a function, each input should correspond with only one output
Hence, from the above,
We can conclude that the given relation is a function

Question 4.

Answer:
The given figure is:

The representation of the given figure in the form of relation is:
(1, -2), (1, 0), (2, 4)
From the given relation,
We know that,
x represents the input
y represents the output
We know that,
For a function, each input should correspond with only one output
But,
When we observe the given relation, there are 2 outputs for a single input
Hence, from the above,
We can conclude that the given relation is not a function

Monitoring Progress

Determine whether the graph represents a function. Explain.

Question 5.

Answer:
The given graph is:

The representation of the points from the graph is:
(0, 3), (1, 3), (2, 3), (3, 3), (4, 3), (5, 3)
From the given relation,
We know that,
x represents the input
y represents the output
We know that,
For a function, each input should correspond with only one output
We have to remember that the inputs must be different but outputs may be the same or different
Hence, from the above,
We can conclude that the given graph is a function

Question 6.

Answer:
The given figure is:

From the graph,
The representation of the points are:
(2, 1), (1, 2), (1, 3), (1, 4), (0.5, 5), (0.5, 6), (3, 0), (4, 1), (4.2, 2), (4.8, 3), (5, 4), (5.1, 5), (5.1, 6)
From the points,
We can observe that the output is different for the same input
Hence, from the above,
We can conclude that the given graph is not a function

Question 7.

Answer:
The given figure is:

From the given graph,
We can observe that for the same value of x, there are different values of y
Where,
x represents the input
y represents the output
We know that,
For a function, each input must be matched with a single output
Hence, from the above,
We can conclude that the given graph is not a function

Question 8.

Answer:
The given figure is:

From the graph,
We can observe that there are multiple outputs for a single input
Hence, from the above,
We can conclude that the given graph is not a function

Monitoring Progress

Find the domain and range of the function represented by the graph.

Question 9.

Answer:
The given graph is:

We know that,
"Domain" is defined as the set of all values present in the x-axis
"Range" is defined as the set of all values present in the y-axis
Hence,
The domain of the given graph is: -2, -1, 0, 1, 2
The range of the given graph is: 1, 2, 3,  4

Question 10.

Answer:
The given graph is:

We know that,
"Domain" is defined as the set of all values present in the x-axis
"Range" is defined as the set of all values present in the y-axis
Hence,
The domain of the given graph is: 1, 2, 3, 4, 5
The range of the given graph is: 0, 1, 2, 3,  4

Monitoring Progress

Question 11.
The function a = -4b + 14 represents the number 'a' of avocados you have left after making b batches of guacamole.
a. Identify the independent and dependent variables.
Answer:
The "Independent variable" represents the input values of a function and can be any value in the domain.
The dependent variable represents the output values of the function and depends on the value of the independent variable
Hence, from the above,
The independent variable is: b (batches of guacamole)
The dependent variable is: a ( The number of avocados )

b. The domain is 0, 1, 2, and 3. What is the range?
Answer:
The given function is:
a = -4b + 14
In the given function,
Input: b
Output: a
We know that,
The "Domain" is defined as the set of all values present in the input or the x-axis
The given Domain is: 0, 1, 2, and 3
So,
To find the range, we have to find the values of b for each value present in the domain
So,
a = -4 (0) + 14 = 14
a = -4 (1) + 14 = 10
a = -4 (2) + 14 = 6
a = -4 (3) + 14 = 2
Hence, from the above,
We can conclude that the range for the given equation is: 2,  6, 10, and 14

Question 12.
The function t = 19m + 65 represents the temperature t (in degrees Fahrenheit) of an oven after preheating for m minutes.
a. Identify the independent and dependent variables.
Answer:
The "Independent variable" represents the input values of a function and can be any value in the domain.
The dependent variable represents the output values of the function and depends on the value of the independent variable
Hence, from the above,
The independent variable is: Minutes
The dependent variable is: Temperature

b. A recipe calls for an oven temperature of 350°F. Describe the domain and range of the function.
Answer:
The given function is:
t = 19m + 65
Compare the given function with
y = mx + c
It is given that a recipe calls for an oven temperature of 350°F
So,
350 = 19m + 65
19m = 350 – 65
19m = 285
m = 285 / 19
m = 15 minutes
So,
t = 19 (15) + 350
t = 285 + 65
t = 350°F
Hence, from the above,
We can conclude that
The domain of the given function is: 0 ≤ m ≤ 15 [ Since the minutes will not be -ve ]
The range of the given function is: 65 ≤ t ≤ 350 [ Since the  minimum temperature is the value of c ]

Functions 3.1 Exercises

Vocabulary and Core Concept Check

Question 1.
WRITING
How are independent variables and dependent variables different?
Answer:

Question 2.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find "both" answers.

Answer:
The given statements are:
a. Find the range of the function represented by the table?
b. Find the inputs of the function represented by the table?
c. Find the x values of the function represented by (-1, 7), (0, 5), and (1, -1)?
d. Find the domain of the function represented by (-1, 7), (0, 5), and (1, -1)?
Now,
From the given table,
The values of x are: -1, 0, 1
The values of y are: 7, 5, -1
We know that,
For a function,
The x values represent the input and the domain
The y values represent the output and the range
So,
a.
The range of the function represented by the table is: 7, 5, -1
b.
The inputs of the function represented by the table are: -1, 0, 1
c.
The x values of the function represented by the given points are: -1, 0, 1
d.
The domain of the function represented by the given points are: -1, 0, 1
Hence, from the above,
We can conclude that all the given four are the same

Monitoring Progress and Modeling with Mathematics

In Exercises 3–8, determine whether the relation is a function. Explain.

Question 3.
(1, -2), (2, 1), (3, 6), (4, 13), (5, 22)
Answer:

Question 4.
(7, 4), (5, -1), (3, -8), (1, -5), (3, 6)
Answer:
The given points are:
(7, 4), (5, -1), (3, -8), (1, -5), (3, 6)
We know that,
For a function, each input has only a single output
But from the above,
We can observe that 3 has multiple outputs
Hence, from the above,
We can conclude that the given relation is not a function

Question 5.

Answer:

Question 6.

Answer:
The given relation is:

We know that,
For a function, a single input has a single output
Hence, from the above,
We can say that each input has only 1 output
Hence, from the above,
We can conclude that the given relation is a function

Question 7.

Answer:

Question 8.

Answer:
The given table is:

From the given table,
We can observe that each input has a single output
Hence, from the above,
We can conclude that the given relation is a function

In Exercises 9–12, determine whether the graph represents a function. Explain.

Question 9.

Answer:

Question 10.

Answer:
The given graph is:

From the given graph,
We can observe that the vertical line can be drawn through more than one point on the graph i.e., input 2 has repeated 2 times i.e., (2, 1) and (2, 5)
Hence, from the above,
We can conclude that the given graph is not a function

Question 11.

Answer:

Question 12.

Answer:
The given graph is:

From the given graph,
We can observe that each input corresponds to a single output
Hence, from the above,
We can conclude that the given graph is a function

In Exercises 13–16, find the domain and range of the function represented by the graph.

Question 13.

Answer:

Question 14.

Answer:
The given graph is:

From the given graph,
The ordered pairs are: (0, 4), (-2, 4), (2, 4), (4, 4)
Hence,
The domain of the given graph is: 0, -2, 2, 4
The range of the given graph is: 4

Question 15.

Answer:

Question 16.

Answer:
The given graph is:

From the given graph,
Identify the x and y values
So,
The values of x range from 2 to 7 excluding 2 and 7
The values of y range from 1 to 6 excluding 1 and 6
Hence, from the above,
We can conclude that
The domain of the given graph is: 2 < x < 7
The range of the given graph is: 1 < y < 6

Question 17.
MODELING WITH MATHEMATICS
The function y = 25x + 500 represents your monthly rent y (in dollars) when you pay x days late.
a. Identify the independent and dependent variables.
b. The domain is 0, 1, 2, 3, 4, and 5. What is the range?
Answer:

Question 18.
MODELING WITH MATHEMATICS
The function y = 3.5x + 2.8 represents the cost y (in dollars) of a taxi ride of x miles.

a. Identify the independent and dependent variables.
Answer:
The given function is:
y = 3.5x + 2.8
From the above function,
The independent variable is: x which represents the number of miles
The dependent variable is: y which represents the cost in dollars

b. You have enough money to travel at most 20 miles in a taxi. Find the domain and range of the function.
Answer:
The given function is:
y = 3.5x + 2.8
It is given that you have enough money to travel at most 20 miles i.e., the value of x in a taxi i.e.,
x ≥ 0 and x ≤ 20
So,
The value of x ranges from 0 ≤ x ≤ 20
Now,
y = 3.5 (20) + 2.8
y = 7 + 2.8
y = $9.8
y = 3.5 (0) + 2.8
y = 0 + 2.8
y = $2.8
Hence, from the above,
We can conclude that
The domain of the given function is: 0 ≤ x ≤ 20 miles
The range of the given function is: $2.8 ≤ y ≤ $9.8

ERROR ANALYSIS
In Exercises 19 and 20, describe and correct the error in the statement about the relation shown in the table.

Question 19.

Answer:

Question 20.

Answer:
It is given that the relation is a function and the range is 1, 2, 3, 4, and 5
We know that,
The relation is a function only when a single input pairs with an output
For the function, the domain, and the range exist
From the given table,
The domain is: 1, 2, 3, 4, 5
The range is: 6, 7, 8, 6, 9
Hence, from the above,
We can conclude that the given statement is not correct

ANALYZING RELATIONSHIPS
In Exercises 21 and 22, identify the independent and dependent variables.

Question 21.
The number of quarters you put into a parking meter affects the amount of time you have on the meter.
Answer:

Question 22.
The battery power remaining on your MP3 player is based on the amount of time you listen to it.
Answer:
The given statement is:
The battery power remaining on your MP3 player is based on the amount of time you listen to it.
Hence, from the above,
We can conclude that
The Independent variable: Amount of time
The dependent variable: Battery power

Question 23.
MULTIPLE REPRESENTATIONS
The balance y (in dollars) of your savings account is a function of the month x.

a. Describe this situation in words.
b. Write the function as a set of ordered pairs.
c. Plot the ordered pairs in a coordinate plane.
Answer:

Question 24.
MULTIPLE REPRESENTATIONS
The function 1.5x + 0.5y = 12 represents the number of hardcover books x and softcover books y you can buy at a used book sale.
a. Solve the equation for y.
Answer:
The given function is:
1.5x + 0.5y = 12
So,
0.5y = 12 – 1.5x
y = \(\frac{12 – 1.5x}{0.5}\)
y = \(\frac{12}{0.5}\) – \(\frac{1.5x}{0.5}\)
y = 24 – 3x
Hence, from the above,
We can conclude that the equation for y is:
y = 24 – 3x

b. Make an input-output table to find ordered pairs for the function.
Answer:
From part (a),
The equation for y is:
y = 24 – 3x
Now,
Put the values of 0, 1, 2, 3 in x
So,
y = 24 – 3(0) = 24
y = 24 – 3(1) = 21
y = 24 – 3(2) = 18
y = 24 – 3 (3) = 15
Hence,
The input-output table for the given equation of y is:

c. Plot the ordered pairs in a coordinate plane.
Answer:
From part (b),
The table is:

We know that,
The ordered pair is in the form of (x, y)
From the table,
The ordered pairs are:
(0, 24), (1, 21), (2, 18), (3, 15)
Hence,
The representation of the ordered pairs in the coordinate plane is:

Question 25.
ATTENDING TO PRECISION
The graph represents a function. Find the input value corresponding to an output of 2.

Answer:

Question 26.
OPEN-ENDED
Fill in the table so that when t is the independent variable, the relation is a function, and when t is the dependent variable, the relation is not a function.

Answer:
Let the function in terms of t and v  such as t is the independent variable and v is the dependent variable is:
v = t + 4 ———–(1)
Let the function in terms of t and v  such as v is the independent variable and t is the dependent variable is:
t = 4v ———–(2)
Now,
place the values 0, 1, 2, 3 …….. in the independent variables of both functions
Now,
In equation (1),
v = 0 + 4 = 4
v = 1 + 4 = 5
v = 2 + 4 = 6
v = 3 + 4 = 7
In equation (2),
t = 4(0) = 0
t = 4(1) = 4
t = 4(2) = 8
t = 4(3) = 12
Hence,

Hence, from the above,
We can conclude that equation (1) is a function and equation (2) is not a function

Question 27.
ANALYZING RELATIONSHIPS
You select items in a vending machine by pressing one letter and then one number.

a. Explain why the relation that pairs letter-number combinations with food or drink items is a function.
b. Identify the independent and dependent variables.
c. Find the domain and range of the function
Answer:

Question 28.
HOW DO YOU SEE IT?
The graph represents the height h of a projectile after t seconds.

a. Explain why h is a function of t.
b. Approximate the height of the projectile after 0.5 seconds and after 1.25 seconds.
c. Approximate the domain of the function.
d. Is t a function of h? Explain.
Answer:
a.
The given graph is:

From the graph,
We can observe that when we draw the vertical lines, each vertical line corresponds to only 1 value.
Hence, from the above,
We can conclude that h is a function of t by using the vertical line method

b.
From the given graph,
We can observe that the height of the projectile after 0.5 seconds increases and after some time, the height of the projectile decreases
We can observe that the height of the projectile after 1.25 seconds decreases steadily
Hence, from the above,
We can conclude that
The approximate maximum height after 0.5 seconds is: 30 feet
The approximate maximum height after 1.25 seconds is: 25 feet

c.
From the given graph,
The values of t in the x-axis vary from 0 to 2.5
Hence,
The domain of the given graph is: 0 ≤ t ≤ 2.5 seconds

d.
t is not a function of h
Reason:
When we observe the graph,
We can see that for a single value of t, there are multiple values of h.
We know that a relation can be considered as a function only when a single input pairs with a single output
hence, from the above,
We can conclude that that t is not a function of h

Question 29.
MAKING AN ARGUMENT
Your friend says that a line always represents a function. Is your friend correct? Explain.
Answer:

Question 30.
THOUGHT-PROVOKING
Write a function in which the inputs and/or the outputs are not numbers. Identify the independent and dependent variables. Then find the domain and range of the function.
Answer:
It is given that the inputs and/or the outputs will not be numbers.
So,
The given function in which the inputs and/or the outputs are not numbers is:
°C = 32 + °F
From the given function,
The independent variable is: °F
The dependent variable is: °C
Now,
From the given function,
We can observe that the values of °F vary from -∞ to ∞
Now,
Place the values of -∞ to ∞ in the place of °F
So,
°C = 32 + 0 = 32°F
°C = 32 + 1 = 33°F
°C = 32 – 1 = 31°F
°C = 32 – 40 = -8°F
Hence, from the above values,
We can observe that the values of °C vary from -∞ to ∞
Hence, from the above,
We can conclude that
The domain of the function is: -∞ to ∞
The range of the function is: -∞ to ∞

ATTENDING TO PRECISION In Exercises 31–34, determine whether the statement uses the word function in a way that is mathematically correct. Explain your reasoning.

Question 31.
The selling price of an item is a function of the cost of making the item.
Answer:

Question 32.
The sales tax on a purchased item in a given state is a function of the selling price.
Answer:
The given statement is:
The sales tax on a purchased item in a given state is a function of the selling price
We know that,
The sales tax is a percentage applied to the selling price
Hence, from the above,
We can conclude that the given statement uses the word function in a way that is mathematically correct.

Question 33.
A function pairs each student in your school with a homeroom teacher.
Answer:

Question 34.
A function pairs each chaperone on a school trip with 10 students.
Answer:
The given statement is:
A function pairs each chaperone on a school trip with 10 students.
We know that,
Each chaperone on a school trip pairs with more than 1 student i.e., the number may be 2 or ∞ but not exactly 10
Hence, from the above,
We can conclude that the given statement does not use the word function in a way that is mathematically correct.

REASONING
In Exercises 35–38, tell whether the statement is true or false. If it is false, explain why.

Question 35.
Every function is a relation.
Answer:

Question 36.
Every relation is a function.
Answer:
The given statement is false

Explanation:
We know that,
Every function is a relation that has only 1 output for a single input
But we can not say that every relation has a single output for a single input
Hence, from the above,
We can conclude that the given statement is false

Question 37.
When you switch the inputs and outputs of any function, the resulting relation is a function.
Answer:

Question 38.
When the domain of a function has an infinite number of values, the range always has an infinite number of values.
Answer:
The given statement is false

Explanation:
We know that,
The domain is defined as the set of all the values of x
The range is defined as the set of all the values of y
Now, consider an example
Let the input be x
Let the output be a constant
Now,
The domain of the input can vary from -∞ to ∞
But the range of the output is only a constant even though we put any value of x
Hence, from the above,
We can conclude that the given statement is false

Question 39.
MATHEMATICAL CONNECTIONS
Consider the triangle shown.

a. Write a function that represents the perimeter of the triangle.
b. Identify the independent and dependent variables.
c. Describe the domain and range of the function. (Hint: The sum of the lengths of any two sides of a triangle is greater than the length of the remaining side.)
Answer:

REASONING
In Exercises 40–43, find the domain and range of the function.

Question 40.
y = | x |
Answer:
The given function is:
y = | x |
We know that,
| x | = x for x > 0
| x | = -x for x < 0
So,
We can put the values of x from -∞ to ∞
So,
The values of x vary from 0 to ∞ since x can't be negative
Hence, from the above,
We can conclude that
The domain of the given function is: -∞ to ∞
The range of the given function is: 0 to ∞

Question 41.
y = – | x |
Answer:

Question 42.
y = | x | – 6
Answer:
The given function is:
y = | x | – 6
We know that,
| x | = x for x > 0
| x | = -x for x < 0
So,
We can vary the values of x from -∞ to ∞
The values of y vary from -6 to ∞
Hence, from the above,
We can conclude that
The domain of the given function is: -∞ to ∞
The range of the given function is: y ≥ -6

Question 43.
y = 4 – | x |
Answer:

Maintaining Mathematical Proficiency

Write the sentence as an inequality. (Section 2.1)

Question 44.
A number y is less than 16.
Answer:
The given worded form is:
A number y is less than 16
Hence,
The representation of the given worded form in the form of inequality is:
y < 16

Question 45.
Three is no less than a number x.
Answer:

Question 46.
Seven is at most the quotient of a number d and -5.
Answer:
The given worded form is:
Seven is at most the quotient of a number d and -5
Hence,
The representation of the given worded form in the form of inequality is:
7 ≤ d ÷ (-5)

Question 47.
The sum of a number w and 4 is more than -12.
Answer:

Evaluate the expression.

Question 48.
11²
Answer:
The product of 11² is:
11² = 11 × 11 = 121

Question 49.
(-3)⁴
Answer:

Question 50.
-5²
Answer:
The product of -5² is:
-5² = -5 × -5 = 25 [ We know that – × – = + ]

Question 51.
2⁵
Answer:

Lesson 3.2 Linear Functions

Essential Question

How can you determine whether a function is linear or non-linear?
Answer:
Simplify the equation as closely as possible to the form of y = mx + c.
Check to see if your equation has exponents.
If it has exponents, it is nonlinear. If your equation has no exponents, it is linear.

EXPLORATION 1
Finding Patterns for Similar Figures
Work with a partner. Copy and complete each table for the sequence of similar figures. (In parts (a) and (b), use the rectangle shown.) Graph the data in each table. Decide whether each pattern is linear or nonlinear. Justify your conclusion.
a. perimeters of similar rectangles

Answer:
We know that,
The perimeter of the rectangle (P) = 2 (Length + Width)
P = 2 (x + 2x)
P = 2 (3x)
P = 6x
Hence,
The completed table for the perimeters of similar rectangles is:

The representation of the perimeters of the similar rectangles in the coordinate plane is:

b. areas of similar rectangles

Answer:
We know that,
Area of the rectangle (A) = Length × Width
A = x × (2x)
A = 2x²
Hence,
The complete table for the area of the similar rectangles is:

The representation of the areas of the similar rectangles in the coordinate plane is:

c. circumferences of circles of radius r
Answer:
We know that,
The circumference of circle = 2πr
Take the value of π as 3
Hence,
The complete table for the circumferences of circles of radius r is:

The representation of the circumferences of the circles in the coordinate plane is:

d. areas of circles of radius r
Answer:
We know that,
The area of the circle = πr²
Take the value of π as 3
Hence,
The complete table for the areas of the similar circles is:

The representation of the areas of the similar circles is:

Communicate Your Answer

Question 2.
How do you know that the patterns you found in Exploration 1 represent functions?
Answer:
In Exploration 1,
From the graphs,
We can observe that from the vertical test, only one point passes through each vertical line i.e., each input hs only 1 output
Our observation coincides with the definition of function
Hence, from the above,
We can conclude that the given patterns in Exploration 1 represent functions

Question 3.
How can you determine whether a function is linear or nonlinear?
Answer:
Simplify the equation as closely as possible to the form of y = mx + c.
Check to see if your equation has exponents.
If it has exponents, it is nonlinear. If your equation has no exponents, it is linear.

Question 4.
Describe two real-life patterns: one that is linear and one that is nonlinear. Use patterns that are different from those described in Exploration 1.
Answer:
The real-life pattern that is linear is:
The distance you travel when you go for a jog, you can graph the function and make some assumptions with only two points. The slope of a function is the same as the rate of change for the dependent variable (y), For instance, if you're graphing distance Vs
The real-life pattern that is non-linear is:
Triangulation of GPS signals
Example:
A device like your cellphone receives signals from GPS satellites, which have known orbital positions around the Earth.

3.2 Lesson

Monitoring Progress

Does the graph or table represent a linear or nonlinear function? Explain.

Question 1.

Answer:
The given graph is:

The graph represents a straight line
We know that,
The straight line will be in the form of
y = mx + c or y = mx
Where,
m is the slope-intercept
c is the y-intercept that cuts through the y-axis
Hence, from the above,
We can conclude that the given graph is linear function

Question 2.

Answer:
The given graph is:

From the given graph,
We can observe that the graph is not a straight line i.e., it is not linear
Hence, from the above,
We can conclude that the given graph is a non-linear function

Question 3.

Answer:
The given table is:

From the given table,
We can observe that
There is a constant difference of 1 between the values of x and there is a constant difference of 2 between the values of y
The difference is constant for both the values of x and y
Hence, from the above,
We can conclude that the given table is a linear function

Question 4.

Answer:
The given table is:

From the given table,
We can observe that
There is a constant difference of 1 between the values of x and there is a constant ratio of 2 between the values of y.
Since the operations are different between the values of x and y,
We can conclude that the given table is a non-linear function

Does the equation represent a linear or nonlinear function? Explain.

Question 5.
y = x + 9
Answer:
The given equation represents the linear function

Explanation:
The given function is:
y = x + 9
We know that,
The standard representation of the linear function is:
y = mx + c
When we compare the given equation with the standard representation of linear function,
We can conclude that the given equation is a linear function

Question 6.
y = \(\frac{3 x}{5}\)
Answer:
The given equation represents a linear function

Explanation:
The given function is:
y = \(\frac{3 x}{5}\)
5y = 3x
3x – 5y = 0
y = \(\frac{3}{5}\)x + 0
We know that,
The standard representation of a linear function is:
y = mx + c
When we compare the above function with the standard representation of a linear function,
We can conclude that the given function is a linear function

Question 7.
y = 5 – 2x²
Answer:
The given equation represents a non-linear function

Explanation:
The given function is:
y = 5 – 2x²
We know that,
The standard representation of a linear function is:
y = mx + c
When we compare the given function with the standard representation of a linear function,
We can conclude that the given function is a non-linear function

Question 8.
The linear function m = 50 – 9d represents the amount m (in dollars) of money you have after buying d DVDs.
(a) Find the domain of the function. Is the domain discrete or continuous? Explain.
Answer:
The given linear function is:
m = 50 – 9d
Where,
m is the amount in dollars you have after buying d DVDs
m is the dependent variable
d is the independent variable
We know that,
The domain is defined for the independent variables
So,
Let
d = 0, 1, 2, 3, 4, 5……
Now,
m = 50 – 9 (0) = 50
m = 50 – 9 (1) = 41
m = 50 – 9 (2) = 32
m = 50 – 9 (3) = 23
m = 50 – 9 (4) = 14
m = 50 – 9 (5) = 5
m = 50 – 9 (6) = -4
Hence, from the above,
We can conclude that
The domain of the given linear function is: 0, 1, 2, 3, 4, and 5 [ Since from 6, -ve values are coming and the money will not be -ve ]
The domain is discrete [ Since a discrete graph is a series of unconnected points ]

(b) Graph the function using its domain.
Answer:
From part (a),
The domain of the given function is: 0, 1, 2, 3, 4, and 5
Hence,
The representation of the domain in the coordinate plane is:

Question 9.
Is the domain discrete or continuous? Explain.

Answer:
The given table is:

From the given table,
The ordered pairs are:
(1, 12), (2, 24), (3, 36)
Hence,
The representation of the ordered pairs in the coordinate plane is:

From the above graph,
We can say that the points are scattered or discrete and they are unconnected
Hence, from the above,
We can conclude that the domain for the given table is discrete

Question 10.
A 20-gallon bathtub is draining at a rate of 2.5 gallons per minute. The number g of gallons remaining is a function of the number m of minutes.
a. Does this situation represent a linear function? Explain.
Answer:
It is given that a 20-gallon bathtub is draining at a rate of 2.5 gallons per minute.
Where,
The number g of gallons remaining is a function of the number m of minutes.
So,
From the above,
We can say that the bathtub is draining at the constant rate
We know that,
From the property of linear function, the change will be constantly increasing or decreasing
So,
The representation of the linear function for this situation is:
g = 20 – 2.5x
Hence, from the above,
We can conclude that the given situation is a linear function

b. Find the domain of the function. Is the domain discrete or continuous? Explain.
Answer:
From part (a),
We can conclude that the given situation is a linear function
So,
The domain of the function = \(\frac{Total volume of the bathtub}{The rate of draining}\)
= \(\frac{20}{2.5}\)
= \(\frac{200}{25}\)
= 8
Hence,
The domain of the function is: 0 ≤ x ≤ 8 [ Since the draining rate will not be -ve ]
From the representation of the domain,
We can conclude that the domain is continuous

c. Graph the function using its domain.
Answer:
From part (a),
The linear function is:
g = 20 – 2.5x
We know that,
The domain is: 0 ≤ x ≤ 8
So,
g = 20 – 2.5 (0) = 20
g = 20 – 2.5 (1) = 17.5
g = 20 – 2.5 (2) = 15
g = 20 – 2.5 (3) = 12.5
g = 20 – 2.5 (4) = 10
g = 20 – 2.5 (5) = 7.5
g = 20 – 2.5 (6) = 5
g = 20 – 2.5 (7) = 2.5
g = 20 – 2.5 (8) = 0
Hence,
The representation of the linear function using the domain in the coordinate plane is:

Write a real-life problem to fit the data shown in the graph. Is the domain of the function discrete or continuous? Explain.

Question 11.

Answer:
The given graph is:

The real-life problem that fits the given graph is:
An escalator moves upwards at a constant rate of 1step/minute.
So,
What is the rate escalator moves upwards after 8 minutes?
From the given graph,
We can say that the points are connected
Hence, from the above,
We can conclude that the domain of the function is continuous

Question 12.

Answer:
The given graph is:

The real-life problem that fits the given graph is:
If one company offers to pay you $450 per week and the other offers $10 per hour, and both ask you to work 40 hours per week, which company is offering the better rate of pay?
From the graph,
We can observe that the points are connected
Hence, from the above,
We can conclude that the domain is continuous

Linear Functions 3.2 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
A linear equation in two variables is an equation that can be written in the form ________, where m and b are constants.
Answer:

Question 2.
VOCABULARY
Compare linear functions and nonlinear functions.
Answer:
Linear Function:
A Linear function is a relation between two variables that produces a straight line when graphed.
Example:
y = 2x + 3
Non-Linear Function:
A non-linear function is a function that does not form a line when graphed.
Example:
y = 6x³

Question 3.
VOCABULARY
Compare discrete domains and continuous domains.
Answer:

Question 4.
WRITING
How can you tell whether a graph shows a discrete domain or a continuous domain?
Answer:
A "Discrete domain" is a set of input values that consists of only certain numbers in an interval.
A "continuous domain" is a set of input values that consists of all numbers in an interval.
Sometimes, the set of points that represent the solutions of an equation are distinct, and other times the points are connected.

Monitoring Progress and Modeling with Mathematics

In Exercises 5–10, determine whether the graph represents a linear or nonlinear function. Explain.

Question 5.

Answer:

Question 6.

Answer:
The given graph is:

From the above graph,
By using the vertical test, there is only one point going through each point and the given graph is a straight line
Hence, from the above,
We can conclude that the given graph is a linear function

Question 7.

Answer:

Question 8.

Answer:
The given graph is:

From the above graph,
By using the vertical test, each line passes through only one point and the given graph is a straight line
Hence, from the above,
We can conclude that the given graph is a linear function

Question 9.

Answer:

Question 10.

Answer:
The given graph is:

From the above graph,
By using the vertical test, each line passes through each point and the given graph is not a straight line
Hence, from the above,
We can conclude that the given graph is a non-linear function

In Exercises 11–14, determine whether the table represents a linear or nonlinear function. Explain.

Question 11.

Answer:

Question 12.

Answer:
The given table is:

From the above table,
The difference between all the values of x is 2 which is a constant through all the values of x
The difference between all the values of y is not constant
Since the difference between all the values of y is not constant
The given table is a non-linear function

Question 13.

Answer:

Question 14.

Answer:
The given table is:

From the above table,
The difference between all the values of x is +1 throughout all the values of x
The difference between all the values of y is -15 throughout all the values of y
Since the differences are constant for all the values of x and y,
The given table is a linear function

ERROR ANALYSIS
In Exercises 15 and 16, describe and correct the error in determining whether the table or graph represents a linear function.

Question 15.

Answer:

Question 16.

Answer:
By using the vertical test in the graph,
We can say that each line passes through one point and the given graph is a straight line
Hence, from the above,
We can conclude that the given graph is a linear function

In Exercises 17–24, determine whether the equation represents a linear or nonlinear function. Explain.

Question 17.
y = x² + 13
Answer:

Question 18.
y = 7 – 3x
Answer:
The given equation is a linear function

Explanation:
The given equation is:
y = 7 – 3x
Compare the given equation with the standard representation of the given linear function
The standard representation of the linear function is:
y = mx + c
Hence, from the above,
We can conclude that the given equation is a linear function

Question 19.

Answer:

Question 20.
y = 4x(8 – x)
Answer:
The given equation is not a linear function

Explanation:
The given equation is:
y = 4x (8 – x)
so,
y = 4x (8) – 4x (x)
y = 32x – 4x²
Compare the above equation with the standard representation of the linear function
The standard representation of the linear function is:
y = mx + c
Hence, from the above,
We can conclude that the given equation is not a linear function

Question 21.
2 + \(\frac{1}{6}\) y = 3x + 4
Answer:

Question 22.
y – x = 2x – \(\frac{2}{3}\)y
Answer:
The given equation is a linear fraction

Explanation:
The given equation is:
y – x = 2x – \(\frac{2}{3}\)y
So,
y + \(\frac{2}{3}\)y = 2x + x
\(\frac{3y}{3}\) + \(\frac{2y}{3}\) = 3x
\(\frac{5}{3}\)y = 3x
y = 3x × \(\frac{3}{5}\)
y = \(\frac{9}{5}\)x + 0
Compare the above equation with the standard representation of the linear function
The standard representation of the linear function is:
y = mx + c
Hence, from the above,
We can conclude that the given equation is a linear function

Question 23.
18x – 2y = 26
Answer:

Question 24.
2x + 3y = 9xy
Answer:
The given equation is not a linear fraction

Explanation:
The given equation is:
2x + 3y = 9xy
2x = 9xy – 3y
2x = y (9x – 3)
y = \(\frac{2}{9x – 3}\)x
Compare the above equation with the standard representation of the linear function
The standard representation of the linear function is:
y = mx + c
Hence, from the above,
We can conclude that the given equation is not a linear function

Question 25.
CLASSIFYING FUNCTIONS
Which of the following equations do not represent linear functions? Explain.
A. 12 = 2x² + 4y²
B. y – x + 3 = x
C. x = 8
D. x = 9 – \(\frac{3}{4}\)y
E. y = \(\frac{5x}{11}\)
F = \(\sqrt{x}\) + 3
Answer:

Question 26.
USING STRUCTURE
Fill in the table so it represents a linear function.

Answer:
The given table is:

From the above table,
we can observe that the difference between all the values of x is 5 which is a constant
Now,
To find the difference between all the values of y which is a constant, we have to use the trial and error method.
Now,
If we add +1 to -1 and continue adding +1 to all the values of y, then
-1 + 1 = 0
0 + 1 = 1
1 + 1 = 2
2 + 1 = 3
But the last value is: 11
Now,
If we add +2 to -1 and continue adding +2 to all the values of y, then
-1 + 2 = 1
1 + 2 = 3
3 + 2 = 5
5 + 2 = 7
But the last value is: 11
Now,
If we add +3 to -1 and continue adding +3 to all the values of y, then
-1 + 3 = 2
2 + 3 = 5
5 + 3 = 8
8 + 3 = 11
The last value is also: 11
Hnece, from the above,
We can conclude that we have to add +3 to make all the values of y constant so that the given table represents a linear function
The completed table is:

In Exercises 27 and 28, find the domain of the function represented by the graph. Determine whether the domain is discrete or continuous. Explain.

Question 27.

Answer:

Question 28.

Answer:
The given graph is:

We know that,
The domain is defined as the range of all the values of x
So,
From the above graph,
The domain is: 0, 1, 2, 3, 4, 5, 6, and 7
Hence, from the above,
We can conclude that the domain of the given graph is: 0, 1, 2, 3, 4, 5, 6, and 7

In Exercises 29–32, determine whether the domain is discrete or continuous. Explain.

Question 29.

Answer:

Question 30.

Answer:
The given table is:

From the given table,
The difference between the values of x is 1 which is constant throughout all of the values of x
The difference between the values of y is 3 which is constant throughout all of the values of y
Hence,
Since the difference is constant for both the values of x and y,
The domain of the given table is continuous

Question 31.

Answer:

Question 32.

Answer:
The given table is:

From the above table,
The difference between all the values of x is 1 which is a constant
The difference between all the values of y is 4 which is a constant
Hence,
Since the difference between all the values of x and y is constant,
The given function is a linear function
The domain of the given function is continuous

ERROR ANALYSIS
In Exercises 33 and 34, describe and correct the error in the statement about the domain. 33. xy214324682.5 is in the domain.

Question 33.

Answer:

Question 34.

Answer:
We know that,
The domain is the range of all the values of x
Now,
From the given graph,
The domain of the given function is: 0, 1, 2, 3, 4, 5, and 6
From the given graph,
We can observe that the domain of the given graph is continuous because there are not any unconnected points in the graph

Question 35.
MODELING WITH MATHEMATICS
The linear function m = 55 – 8.5b represents the amount m (in dollars) of money that you have after buying b books.
a. Find the domain of the function. Is the domain discrete or continuous? Explain.
b. Graph the function using its domain.

Answer:

Question 36.
MODELING WITH MATHEMATICS
The number y of calories burned after x hours of rock climbing is represented by the linear function y = 650x.
a. Find the domain of the function. Is the domain discrete or continuous? Explain.
b. Graph the function using its domain.

Answer:
a.
It is given that
The number y of calories burned after x hours of rock climbing is represented by the linear function
y = 650x.
It is given that x is the number of hours
Hence,
The domain of the given function is:
1 ≤ x ≤ 24
Since, the domain of the function is inequality,
The domain of the function is continuous

b.
The given function is:
y = 650x
From part (a),
The domain of the function is: 1 ≤ x ≤ 24
So,
y = 650 (1) = 650
y = 650 (2) = 1300
y = 650 (3) = 1950
.
.
.
.
y = 650 (12) = 7800
Hence,
The representation of the function with the domain in the coordinate plane is:

Question 37.
MODELING WITH MATHEMATICS
You are researching the speed of sound waves in dry air at 86°F. The table shows the distances d (in miles) sound waves travel in t seconds.

a. Does this situation represent a linear function? Explain.
b. Find the domain of the function. Is the domain discrete or continuous? Explain.
c. Graph the function using its domain.
Answer:

Question 38.
MODELING WITH MATHEMATICS
The function y = 30 + 5x represents the cost y (in dollars) of having your dog groomed and buying x extra services.

a. Does this situation represent a linear function? Explain.
Answer:
The given function is:
y = 30 + 5x
We know that,
The standard representation of the linear function is:
y = mx + c
Compare the given function with the standard representation
Hence, from the above,
We can conclude that the given situation represents a linear function

b. Find the domain of the function. Is the domain discrete or continuous? Explain.
Answer:
The given function is:
y = 30 + 5x
Where,
y is the amount in dollars
x is the cost of extra grooming services
From the above,
The maximum number of given extra grooming services is: 5
So,
We can use extra grooming service or not
Hence, from the above,
We can conclude that
The domain of the given function is: 0 ≤ x ≤ 5
Hence,
The domain of the given function is continuous

c. Graph the function using its domain.
Answer:
The given function is:
y = 30 + 5x
We know that,
The domain of the function is: 0 ≤ x ≤ 5
So,
y = 30 + 5(0) = 30
y = 30 + 5 (1) = 35
y = 30 + 5(2) = 40
y = 30 + 5 (3) = 45
y = 30 + 5 (4) = 50
y = 30 + 5 (5) = 55
Hence,
The representation of the function using its domain in the coordinate plane is:

WRITING In Exercises 39–42, write a real-life problem to fit the data shown in the graph. Determine whether the domain of the function is discrete or continuous. Explain.

Question 39.

Answer:

Question 40.

Answer:
The given graph is:

From the above graph,
The real-life situation is:
The temperature of a country in the winter season falls by 2°C
The domain of the graph is continuous as there are not any unconnected points

Question 41.

Answer:

Question 42.

Answer:
The given graph is:

From the above graph,
The real-life situation is:
The number of ants in a colony increase by 2 times per day. So, the number of ants increases by how many times in 5 days?
The domain of the given graph is continuous since there is not any unconnected point in the graph

Question 43.
USING STRUCTURE
The table shows your earnings y (in dollars) for working x hours.
a. What is the missing y-value that makes the table represent a linear function?
b. What is your hourly pay rate?

Answer:

Question 44.
MAKING AN ARGUMENT
The linear function d = 50t represents the distance d (in miles) Car A is from a car rental store after t hours. The table shows the distances Car B is from the rental store.

a. Does the table represent a linear or nonlinear function? Explain.
Answer:
The given table is:

From the above table,
The values of x increases at a constant rate of 2 but the value of y increases by 120 1st time and by 130 2nd time
Hence,
We can observe that for the constant difference of the values of x, there is no constant difference of the values of y
Hence, from the above,
We can conclude that the given table is a linear function

b. Your friend claims Car B is moving at a faster rate. Is your friend correct? Explain.
Answer:
It is given that
The function represented by car A is:
d = 50t
The function represented by car B is:

From the above functions,
We can say that the distance traveled by car A increases at a constant rate whereas the distance traveled by car B increases at a random rate
Hence, from the above
We can conclude that the car B is moving at a faster rate when we observe the above table

MATHEMATICAL CONNECTIONS
In Exercises 45–48, tell whether the volume of the solid is a linear or nonlinear function of the missing dimension(s). Explain.

Question 45.

Answer:

Question 46.

Answer:
The given figure is:

From the above figure,
We can observe that the given figure is a prism
We know that,
The volume of a prism = Area × height
= Length × Width × Height
So,
The volume of a prism (V) =3 × b × 4
V = 12b
Compare the above volume with y = mx + c
So,
V = 12b + 0
Hence, from the above,
We can conclude that the equation represents a Linear function

Question 47.

Answer:

Question 48.

Answer:
The given figure is:

From the above figure,
We can observe that the given figure is a cone
We know that,
The volume of a cone = \(\frac{1}{3}\) πr²h
So,
The volume of a cone (V) = \(\frac{1}{3}\) × \(\frac{22}{7}\) × r² × 15
V = 770r²
Compare the above equation with
y = mx + c
But, the given equation is not in the form of y = mx + c
Hence, from the above,
We can conclude that the equation represents a non-linear function

Question 49.
REASONING
A water company fills two different-sized jugs. The first jug can hold x gallons of water. The second jug can hold y gallons of water. The company fills A jugs of the first size and B jugs of the second size. What does each expression represent? Does each expression represent a set of discrete or continuous values?
a. x + y
b. A + B
c. Ax
d. Ax + By

Answer:

Question 50.
THOUGHT-PROVOKING
You go to a farmer's market to buy tomatoes. Graph a function that represents the cost of buying tomatoes. Explain your reasoning.
Answer:
It is given that you go to a farmer's market to buy tomatoes.
So,
To draw the graph that represents the cost of buying tomatoes,
The required relation is:
Cost of tomatoes ∝ Quantity or weight of tomatoes
We know that,
∝ represents the direct relation. In a graph, this relation can be represented in a straight line
Hence,
The representation of the relation of cost of buying tomatoes and weight of tomatoes is:

Question 51.
CLASSIFYING A FUNCTION
Is the function represented by the ordered pairs linear or nonlinear? Explain your reasoning.
(0, 2), (3, 14), (5, 22), (9, 38), (11, 46)
Answer:

Question 52.
HOW DO YOU SEE IT?
You and your friend go running. The graph shows the distances you and your friend run.

a. Describe your run and your friend's run. Who runs at a constant rate? How do you know? Why might a person not run at a constant rate?
Answer:
The given graph is:

From the graph,
The running represented by you is a straight line
The running represented by your friend is not a straight line
We know that,
A straight line has a constant rate
Hence, from the above,
We can conclude that you run at a constant rate and your friend does not run at a constant rate

b. Find the domain of each function. Describe the domains using the context of the problem.
Answer:
From part (a),
The given graph is:

We know that,
The domain is defined as the range of the values of x
Hence, from the above,
We can conclude that
The domain of the function related to you is: 0 ≤ x ≤ 50
The domain of the function related to your friend is: 0 ≤ x ≤ 50

WRITING
In Exercises 53 and 54, describe a real-life situation for the constraints.

Question 53.
The function has at least one negative number in the domain. The domain is continuous.
Answer:

Question 54.
The function gives at least one negative number as an output. The domain is discrete.
Answer:
When you go on a world tour and for some days, you stayed in Antarctica,
The temperatures in Antarctica is at a negative temperature around the year and only for some months in the year, the temperature will be positive

Maintaining Mathematical Proficiency

Tell whether x and y show direct variation. Explain your reasoning.

Question 55.

Answer:

Question 56.

Answer:
The given graph is:

From the graph,
We can observe that it is a straight line and passes through the origin
Hence, from the above,
We can conclude that x and y shows direct variation

Question 57.

Answer:

Evaluate the expression when x = 2.

Question 58.
6x + 8
Answer:
The given expression is: 6x + 8
When x = 2,
6x + 8 = 6 (2) + 8
= 12 + 8 = 20
Hence,
The value of the expression when x = 2 is: 20

Question 59.
10 – 2x + 8
Answer:

Question 60.
4(x + 2 – 5x)
Answer:
The given expression is: 4 (x + 2 – 5x)
When x = 2,
4 (x + 2 – 5x) = 4 (2 + 2 – 5 (2) )
= 4 (4 – 10 )
= 4 (-6) = -24
Hence,
The value of the expression when x = 2 is: -24

Question 61.
\(\frac{x}{2}\) + 5x – 7
Answer:

Lesson 3.3 Function Notation

Essential Question

How can you use function notation to represent a function?
The notation f(x), called function notation, is another name for y. This notation is read as "the value of f at x" or "f of x." The parentheses do not imply multiplication. You can use letters other than f to name a function. The letters g, h, j, and k are often used to name functions.

EXPLORATION 1
Matching Functions with Their Graphs
Work with a partner. Match each function with its graph.
a. f (x) = 2x – 3
b. g(x) = -x + 2
c. h(x) = x² – 1
d. j(x) = 2x² – 3


Answer:
The given equations are:
a. f (x) = 2x – 3
b. g(x) = -x + 2
c. h(x) = x² – 1
d. j(x) = 2x² – 3
Now,
a.
The given equation is:
f(x) = 2x – 3
So,
The representation of the given equation in the coordinate plane is:

Hence, from the above,
We can conclude that graph B) matches this equation
b.
The given equation is:
f(x) = 2x – 3
So,
The representation of the given equation in the coordinate plane is:

Hence, from the above,
We can conclude that graph D) matches this equation
c.
The given equation is:
h(x) = x² – 1
So,
The representation of the given equation in the coordinate plane is:

Hence, from the above,
We can conclude that graph A) matches this equation
d.
The given equation is:
j(x) = 2x² – 3
So,
The representation of the given equation in the coordinate plane is:

Hence, from the above,
We can conclude that graph C) matches this equation

EXPLORATION 2
Evaluating a Function
Work with a partner. Consider the function
f(x) = -x + 3.
Locate the points (x, f(x)) on the graph. Explain how you found each point.

a. (-1, f(-1))
b. (0, f(0))
c. (1, f(1))
d. (2, f(2))
Answer:
The given function is:
f(x) = -x + 3
The graph for the given function is:

Now,
The simplified points are:
a.
(-1, f(-1)) = (-1, [-(-1) + 3]) = (-1, 4)
b.
(0, f(0)) = (0, [0 + 3]) = (0, 3)
c.
(1, f(1)) = (1, [-1 + 3]) = (1, 2)
d.
(2, f(2)) = (2, [-2 + 3]) = (2, 1)
So,
The simplified points are:
(-1, 4), (0, 3), (1, 2), (2, 1)
Hence,
The representation of the above points in the graph is:

Communicate Your Answer

Question 3.
How can you use function notation to represent a function? How are standard notation and function notation similar? How are they different?

Answer:
"Function notation" is a simpler method of describing a function without a lengthy written explanation. The most frequently used function notation is f(x) which is read as "f" of "x".
The standard notation and function notation are similar in the way of simplification
Difference betwwen Function notataion and standard notation:
The representation of function notation largely depends on the number of variables present in the function
Ex:
f(x,y) = 2xy + 3
f(x) = x + 3
The representation of the standard notation does not depend on the number of variables present in the equation.

3.3 Lesson

Monitoring Progress

Evaluate the function when x = −4, 0, and 3.

Question 1.
f(x) = 2x – 5
Answer:
The given function is:
f(x) = 2x – 5
Now,
When x = -4,
f(-4) = 2 (-4) – 5 = -8 – 5 = -13
When x = 0,
f(0) = 2 (0) – 5 = 0 – 5 = -5
When x = 3,
f(3) = 2 (3) – 5 = 6 – 5 = 1
Hence, from the above,
We can conclude that the values of f(x) when x = -4, 0, 3 are: -13, -5, and 1

Question 2.
g(x) = -x – 1
Answer:
The given function is:
g(x) = -x – 1
Now,
When x = -4,
g(-4) = -[-4] – 1 = 4 – 1 = 3
When x = 0,
g(0) = 0 – 1 = -1
When x = 3,
g(3) = -3 – 1 = -4
Hence, from the above,
We can conclude that the values of g(x) when x = -4, 0, 3 are: 3, -1, and -4

Question 3.
WHAT IF? In Example 2, let f(t) be the outside temperature (°F) t hours after 9 A.M. Explain the meaning of each statement.
a. f(4) = 75
b. f(m) = 70
c. f(2) = f(9)
d. f(6) > f(0)

Monitoring Progress

Find the value of x so that the function has the given value.

Question 4.
f(x) = 6x + 9; f(x) = 21
Answer:
The value of x is: 2

Explanation:
The give function is:
f(x) = 6x + 9 with f(x) = 21
So,
21 = 6x + 9
6x = 21 – 9
6x = 12
x = 12 / 6
x = 2
Hence, from the above,
We can conclude that the value of x in the given function is: 2

Question 5.
g(x) = \(-\frac{1}{2}\)x + 3; g(x) = -1
Answer:
The value of x is: 8

Explanation:
The given function is:
g(x) = \(-\frac{1}{2}\)x + 3 with g(x) = -1
So,
-1 = \(-\frac{1}{2}\)x + 3
\(-\frac{1}{2}\)x = -1 – 3
\(-\frac{1}{2}\)x = -4
–\(\frac{1}{2}\)x = -4
\(\frac{1}{2}\)x = 4
x = 2(4)
x = 8
Hence, from the above,
We can conclude that the value of x in the given function is: 8

Graph the linear function.

Question 6.
f(x) = 3x – 2
Answer:
The given function is:
f(x) = 3x – 2
Now,
put the values -2, -1, 0, 1, 2 in the place of x and find the values of f(x) toplot a graph [ Remember you can take any value and any number of values]
So,
f(-2) = 3 (-2) – 2 = -6 – 2 = -8
f(-1) = 3(-1) – 2 = -3 – 2 = -5
So,
The completed table for the given function is:

Hence,
The representation of the given function in the coordinate plane is:

Question 7.
g(x) = -x + 4
Answer:
The given function is:
g(x) = -x + 4
Now,
put the values -2, -1, 0, 1, 2 in the place of x and find the values of f(x) toplot a graph [ Remember you can take any value and any number of values]
So,
g(-2) = -[-2] + 4 = 2 + 4 = 6
g(-1) = -[-1] + 4 = 4 + 1 = 5
So,
The completed table for the given function is:

Hence,
The representation of the given function in the coordinate plane is:

Question 8.
h(x) = \(-\frac{3}{4}\)x – 1
Answer:
The given function is:
h(x) = \(-\frac{3}{4}\)x – 1
Now,
put the values -2, -1, 0, 1, 2 in the place of x and find the values of f(x) toplot a graph [ Remember you can take any value and any number of values]
So,
h(-2) = \(-\frac{3}{4}\) (-2) – 1
= \(\frac{3}{2}\) – 1
= \(\frac{1}{2}\)
h(-1) = \(-\frac{3}{4}\) (-1) – 1
= \(\frac{3}{4}\) – 1
= \(-\frac{1}{4}\)
So,
The completed table for the given function is:

Hence,
The representation of the given function in the coordinate plane is:

Question 9.
WHAT IF?
Let f(x) = 250 – 75x represent the second flight, where f(x) is the number of miles the helicopter is from its destination after x hours. Which flight takes less time? Explain.

Function Notation 3.3 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
When you write the function y = 2x + 10 as f(x) = 2x + 10, you are using ______________.
Answer:

Question 2.
REASONING
Your height can be represented by a function h, where the input is your age. What does h(14) represent?
Answer:
It is given that your height can be represented by a function h, where the input is your age
So,
h (14) means you are 14 years old

Monitoring Progress and Modeling with Mathematics

In Exercises 3–10, evaluate the function when x = –2, 0, and 5.

Question 3.
f(x) = x + 6
Answer:

Question 4.
g(x) = 3x
Answer:
The given function is:
g (x) = 3x
When x = -2,
g (-2) = 3 (-2) = -6
When x = 0,
g (0) = 3 (0) = 0
When x = 5,
g (5) = 3 (5) = 15
Hence, from the above,
We can conclude that the values of g (x) when x = -2, 0, 5 is: -6, 0, and 15

Question 5.
h(x) = -2x + 9
Answer:

Question 6.
r(x) = -x – 7
Answer:
The given function is:
r (x) = -x – 7
When x = -2,
r (-2) = -[-2] – 7
= 2 – 7 = -5
When x = 0,
r (0) = 0 – 7 = -7
When x = 5,
r (5) = -5 – 7 = -12
Hence, from the above,
We can conclude taht the values of r (x) when x = 0, -2, 5 is: -5, -7, and -12

Question 7.
p(x) = -3 + 4x
Answer:

Question 8.
b(x) = 18 – 0.5x
Answer:
The given function is:
b (x) = 18 – 0.5x
When x = -2,
b (-2) = 18 – 0.5 (-2)
= 18 + 1 = 19
When x = 0,
b (0) = 18 – 0.5 (0) = 18
When x = 5,
b (5) = 18 – 0.5 (5)
18 – 2.5 = 15.5
Hence, from the above,
We can conclude that the values of b (x) when x = 0, -2, 5 is: 19, 18, and 15.5

Question 9.
v(x) = 12 – 2x – 5
Answer:

Question 10.
n(x) = -1 – x + 4
Answer:
The given function is:
n (x) = -1 – x + 4
When x = -2,
n (-2) = -1 – [-2] + 4
= -1 + 2 + 4
= 6 – 1
= 5
When x = 0,
n (0) = -1 – 0 + 4
= -1 + 4 = 3
When x = 2,
n (2) = -1 – 2 + 4
= 4 – 3
= 1
Hence, from the above,
We can conclude that the values of n (x) when x = -2, 0, 5 is: 5, 3, and 1

Question 11.
INTERPRETING FUNCTION NOTATION
Let c(t) be the number of customers in a restaurant t hours after 8 A.M. Explain the meaning of each statement.
a. c(0) = 0
b. c(3) = c(8)
c. c(n) = 29
d. c(13) < c(12)
Answer:

Question 12.
INTERPRETING FUNCTION NOTATION
Let H(x) be the percent of U.S. households with Internet use x years after 1980. Explain the meaning of each statement.
a. H(23) = 55
b. H(4) = k
c. H(27) ≥ 61
d. H(17) + H(21) ≈ H(29)
Answer:
It is given that H (x) is the percent of U.S households with internet use x years after 1980
Now,
a.
The given function is:
H (23) = 55
The meaning of the above function is:
23 years after 1980, 55% of U.S households will be using the internet

b.
The given function is:
H(4) = k
The meaning of the above function is:
4 years after 1980, k% of U.S households will be using the internet

c.
The given function is:
H(27) ≥ 61
The meaning of the above function is:
27 years after 1980, more than 61% of U.S households will be using the internet

d.
The given function is:
H(17) + H(21) ≈ H(29)
The meaning of the above function is:
The percentage of U.S households using the internet after 29 years is equal to the sum of percentage of household using the internet after 17 years and 21 years

In Exercises 13–18, find the value of x so that the function has the given value.

Question 13.
h(x) = -7x; h(x) = 63
Answer:

Question 14.
t(x) = 3x; t(x) = 24
Answer:
The given function is:
t (x) = 3x with t (x) = 24
So,
24 = 3x
x = 24 / 3
x = 8
Hence, from the above,
We can conclude that the value of the given function is: 8

Question 15.
m(x) = 4x + 15; m(x) = 7
Answer:

Question 16.
k(x) = 6x – 12; k(x) = 18
Answer:
The given function is:
k (x) = 6x – 12 with k (x) = 18
So,
6x – 12 = 18
6x = 12 + 18
6x = 30
x = 30 / 6
x = 5
Hence, from the above,
We can conclude that the value of the given function is: 5

Question 17.
q(x) = \(\frac{1}{2}\)x – 3; q(x) = -4
Answer:

Question 18.
j(x) = –\(\frac{4}{5}\)x + 7; j(x) = -5
Answer:
The given function is:
j (x) = —\(\frac{4}{5}\)x + 7 with j (x) = -5
So,
-5 = –\(\frac{4}{5}\)x + 7
-5 – 7 = –\(\frac{4}{5}\)x
–\(\frac{4}{5}\)x = -12
\(\frac{4}{5}\)x = 12
x = 12 × –\(\frac{5}{4}\)
x = 15
Hence, from the above,
We can conclude that the value of the given function is: 15

In Exercises 19 and 20, find the value of x so that f(x) = 7.

Question 19.

Answer:

Question 20.

Answer:
The given function is:
f (x) = 7
So,
We know that,
y = f (x)
So,
From the above,
We can say that the value of y is: 7
Now,
To find the value of x, observe the location corresponding to the value of y
So,
The value of x is: -2
Hence,
The point of the graph where f (x) = 7 is: (-2, 7)

Question 21.
MODELING WITH MATHEMATICS
The function C(x) = 17.5x – 10 represents the cost (in dollars) of buying x tickets to the orchestra with a $10 coupon.
a. How much does it cost to buy five tickets?
b. How many tickets can you buy for $130?
Answer:

Question 22.
MODELING WITH MATHEMATICS
The function d(t) = 300,000t represents the distance (in kilometers) that light travels in t seconds.
a. How far does light travel in 15 seconds?
b. How long does it take light to travel 12 million kilometers?

Answer:
The given function is:
d (t) = 300,000t
represents the distance in kilometers that travels in t seconds
a.
The distance traveled by light in 15 seconds is:
d (15) = 300,000 (15)
= 4,500,000 kilometers
Hence, from the above,
We can conclude that the distance traveled by light in 15 seconds is: 4,500,000 kilometers

b.
It is given that the total distance traveled by light is 12 million kilometers
So,
d (t) = 12 million kilometers
So,
12,000,000 = 300,000 (t)
t = 12,000,000 / 300,000
t = 40 seconds
Hence, from the above,
We can conclude that the time taken by light to travel 12 million kilometers is: 40 seconds

In Exercises 23–28, graph the linear function.

Question 23.
p(x) = 4x
Answer:

Question 24.
h(x) = -5
Answer:
The given function is:
h (x) = -5
Now,
put the values -2, -1, 0, 1, 2 in the place of x and find the values of f(x) to plot a graph [ Remember you can take any value and any number of values]
So,
The completed table for the given function is:

Hence,
The representation of the given function in the coordinate plane is:

Question 25.
d(x) = \(-\frac{1}{2} x\) – 3
Answer:

Question 26.
w(x) = \(\frac{3}{5} x\) + 2
Answer:
The given function is:
w (x) = \(\frac{3}{5} x\) + 2
So,
w (x) = \(\frac{3}{5}\) x + 2
Now,
put the values -2, -1, 0, 1, 2 in the place of x and find the values of f(x) to plot a graph [ Remember you can take any value and any number of values]
So,
The completed table for the given function is:

Hence,
The representation of the given function in the coordinate plane is:

Question 27.
g(x) = -4 + 7x
Answer:

Question 28.
f(x) = 3 – 6x
Answer:
The given function is:
f (x) = 3 – 6x
Now,
put the values -2, -1, 0, 1, 2 in the place of x and find the values of f(x) to plot a graph [ Remember you can take any value and any number of values]
So,
The completed table for the given function is:

Hence,
The representation of the given function in the coordinate plane is:

Question 29.
PROBLEM-SOLVING
The graph shows the percent p(in decimal form) of battery power remaining in a laptop computer after t hours of use. A tablet computer initially has 75% of its battery power remaining and loses 12.5% per hour. Which computer's battery will last longer? Explain.

Answer:

Question 30.
PROBLEM-SOLVING
The function C(x) = 25x + 50 represents the labor cost (in dollars) for Certified Remodeling to build a deck, where x is the number of hours of labor. The table shows sample labor costs from its main competitor, Master Remodeling. The deck is estimated to take 8 hours of labor. Which company would you hire? Explain.

Answer:
It is given that the given function
C(x) = 25x + 50
represents the labor cost (in dollars) for certified Remodeling to build a deck
Where,
x is the number of hours of labor.
The given table is:

The given table shows the sample labor costs from its main competitor, Master Remodeling.
It is also given that the deck is estimated to take 8 hours of labor.
Now,
From the given table,
We can observe that the competitor 'Master Remodeling' completes the deck in 6 hours whereas for certified Remodeling, it will take 8 hours of labor to complete a deck thereby increases the cost. i.e., the avlue of C(x) also increases when compared to the Master Remodeling
Hence, from the above,
We can conclude that we choose "Master Remodeling" company

Question 31.
MAKING AN ARGUMENT
Let P(x) be the number of people in the U.S. who own a cell phone x years after 1990. Your friend says that P(x + 1) > P(x) for any x because x + 1 is always greater than x. Is your friend correct? Explain.
Answer:

Question 32.
THOUGHT-PROVOKING
Let B(t) be your bank account balance after t days. Describe a situation in which B(0) < B(4) < B(2).
Answer:
The given relation is:
B (0) < B (4) < B (2)
Let us consider
Sunday – Day 0
Monday -Day 1
Tuesday -Day 2
Wednesday – Day 3
Thursday – Day 4
Friday – Day 5
Saturday – Day 6
Now,
By using the above relation,
The situation we can assume is:
A man named A works at a company where he receives his salary every Tuesday (Day 2) of a normal week. He then spends some of his salary paying bills on Thursday (Day 4). On Sunday, he decides to spend all of his remaining salary on food, groceries, and transportation allowance for the following week

Question 33.
MATHEMATICAL CONNECTIONS
Rewrite each geometry formula using function notation. Evaluate each function when r = 5 feet. Then explain the meaning of the result.
a. Diameter, d = 2r
b. Area, A = πr²
c. Circumference, C = 2πr

Answer:

Question 34.
HOW DO YOU SEE IT?
The function y = A(x) represents the attendance at a high school x weeks after a flu outbreak. The graph of the function is shown.

a. What happens to the school's attendance after the flu outbreak?
Answer:
The given graph is:

From the above graph,
We can observe that after the flu outbreak, the school's attendance first decreased and then again steadily increased

b. Estimate A(13) and explain its meaning.
Answer:
The given graph is:

Now,
We know that
The function notation and the standard notations are similar
So,
y = A (x)
So,
y = A (13)
So,
The value of x is: 13
To find the value of y or the given function, observe the graph for the value of y corresponding to the value of x i.e., 13
So,
From the graph,
y = 430 [Approximately]
Hence, from the above,
We can conclude that
The estimation of A(13) is:
A (13) = 430 {Approximately]

c. Use the graph to estimate the solution(s) of the equation A(x) = 400. Explain the meaning of the solution(s).
Answer:
The given graph is:

The given equation is:
A (x) = 400
From the above graph,
For the solution of the given equation i.e., the value of x,
Observe the location of 400 in the graph and its corresponding x-value
Hence,
From the graph,
We can observe that the value of x is: 1
Hence, from the above,
We can conclude that the solution of the given equation is: 1

d. What was the least attendance? When did that occur?
Answer:
The given graph is:

From the above graph,
We can observe that there is deep depreciation in the attendance i.e., least attendance of the students in a week
So,
By observing the graph,
We can say that
The least attendance of the students occurs in the 4th week
The least attendance of the students is: 350
Hence, from the above,
We can conclude that
The week that has the least attendance is: 4th week
The number of students of the least attendance is: 350students

e. How many students do you think are enrolled at this high school? Explain your reasoning.
Answer:
The given graph is:

From the given graph,
We can observe that the y-axis represents the number of students enrolled in the high school
So,
The highest number on the y-axis represents the total number of students enrolled in the high school
Hence, from the above,
We can conclude that the total number of students in the high school is: 450 students

Question 35.
INTERPRETING FUNCTION NOTATION
Let f be a function. Use each statement to find the coordinates of a point on the graph of f.
a. f(5) is equal to 9.
b. A solution of the equation f(n) = -3 is 5.
Answer:

Question 36.
REASONING
Given a function f, tell whether the statement f(a + b) = f(a) + f(b) is true or false for all inputs a and b. If it is false, explain why.
Answer:
Let the given function is:
f or f(x) = mx + c
Now,
Let the values of a and b be integers
So,
Let,
a = 0 and b = 1
So,
f( a + b ) = f (1)
So,
f(1) = m + c
Now,
f(a) = f (0) = c
f (b) = f(1) = m + c
Hence, from the above,
We can conclude that
f (a + b) is not equal to f (a) + f(b)

Maintaining Mathematical Proficiency

Solve the inequality. Graph the solution. (Section 2.5)

Question 37.
-2 ≤ x – 11 ≤ 6
Answer:

Question 38.
5a < -35 or a – 14 > 1
Answer:
The given inequality is:
5a < -35 or a – 14 > 1
So,
a < -35 / 5 or a > 1 + 14
a < -7 or a > 15
Hence,
The solutions of the given inequality are:
a < -7 or a > 15
The representation of the solution of the given inequality in the graph is:

Question 39.
-16 < 6k + 2 < 0
Answer:

Question 40.
2d + 7 < -9 or 4d – 1 > -3
Answer:
The given inequality is:
2d + 7 < -9 or 4d – 1 > -3
2d < -9 – 7 or 4d > -3 + 1
2d < -16 or 4d > -2
d < -16 / 2 or d > -2 / 4
d < -8 or d > -1 /  2
Hence,
The solutions of the given inequality are:
d < -8 or d > -1 / 2
The representation of the solutions of the given inequality in the graph is:

Question 41.
5 ≤ 3y + 8 < 17
Answer:

Question 42.
4v + 9 ≤ 5 or -3v ≥ -6
Answer:
The given inequality is:
4v + 9 ≤5 or -3v ≥ -6
So,
4v ≤ 5 – 9 or 3v ≥ 6
4v ≤ -4 or v ≥ 6 / 3
v ≤ -4 / 4 or v ≥ 2
v ≤ -1 or v ≥ 2
Hence,
The solutions of the given inequality are:
v ≤ -1 or v ≥ 2
The representation of the solutions of the given inequality is:

Graphing Linear Functions Study Skills: Staying Focused During Class

Core Vocabulary

Core Concepts
Section 3.1

Section 3.2

Section 3.3
Using FunctionNotation, p. 122

Mathematical Practices

Question 1.
How can you use technology to confirm your answers in Exercises 40–43 on page 110?
Answer:
We know that,
| x | = x for x > 0
| x | = -x for x < 0
So,
By using the above properties,
We can find the domain and the range of the given Exercises
Hence,
We can confirm the answers in Exercises 40 – 43 on page 110

Question 2.
How can you use patterns to solve Exercise 43 on page 119?
Answer:
In Exercise 43 on page 119,
We can observe from the table that the difference between the values of x and y is constant
So,
By using the above property, we can find the constant difference between the values of x and y to complete the pattern

Question 3.
How can you make sense of the quantities in the function in Exercise 21 on page 125?
Answer:
In Exercise 21 on page 125,
Compare the given function with the standard representation of the linear function y = mx + c
So,
In the above function,
We know that,
m is the cost of x tickets in dollars
c is the constant
f (x) or y is the function notation of Cost function

Study Skills

Staying Focused during Class

As soon as class starts, quickly review your notes from the previous class and start thinking about math.
Repeat what you are writing in your head.
When a particular topic is difficult, ask for another example.

Graphing Linear Functions 3.1 – 3.3 Quiz

Determine whether the relation is a function. Explain. (Section 3.1)

Question 1.

Answer:
The given table is:

We know that for a relationship to be a function,
Each input value has to match with each output value
Hence,
From the above table,
We can conclude that the given table is a function since each input value matches with each output value

Question 2.
(-10, 2), (-8, 3), (-6, 5), (-8, 8), (-10, 6)
Answer:
The given ordered pairs are:
(-10, 2), (-8, 3), (-6, 5), (-8, 8), (-10, 6)
We know that,
A relation is said to be a function if each input matches with only 1 output
So,
By observing the above-ordered pairs,
We can say that the input -8 comes 2 times
Hence, from the above,
We can conclude that the given ordered pairs do not represent a function

Find the domain and range of the function represented by the graph. (Section 3.1)

Question 3.

Answer:
The given graph is:

We know that,
The domain is the set of all x-values
The range is the set of all y-values
So,
The domain of the given graph is: {0, 1, 2, 3, 4}
The range of the given graph is: {1, -1, -3}

Question 4.

Answer:
The given graph is:

We know that,
The domain is the set of all x-values
The range is the set of all y-values
So,
The domain of the given graph is: {-1, -2, 1, 2}
The range of the given graph is: {-2, -1, 0, 1, 2}

Question 5.

Answer:
The given graph is:

We know that,
The domain is the set of all x-values
The range is the set of all y-values
So,
The domain of the given graph is: {-3, -2, -1, 1, 2, 3}
The range of the given graph is: {-1, 0, 1, 2, 3}

Determine whether the graph, table, or equation represents a linear or nonlinear function. Explain. (Section 3.2)

Question 6.

Answer:
The given graph is:

We know that,
A linear function must always represent a straight line irrespective of the straight line passes through the origin or passes through any other point
Hence, from the above,
We can conclude that the given graph is a linear function

Question 7.

Answer:
The given table is:

The representation of the values of x and y in the table in the form of ordered pairs is:
(-5, 3), (0, 7), and (5, 10)
Represent the ordered pairs in the coordinate plane
The representation of the ordered pairs in the coordinate plane is:

From the above points, we can say it forms a straight line
A linear function must always represent a straight line irrespective of the straight line passes through the origin or passes through any other point
Hence, from the above,
We can conclude that the given table represents a linear function

Question 8.
y = x(2 – x)
Answer:
The given function is:
y = x (2 – x)
So,
y = 2 (x) – x (x)
y = 2x – x²
Compare the above function with the standard representation of the linear function
The standard representation of the linear function is:
y = mx + c
Hence, from the above comparison,
We can conclude that the given function is a non-linear function

Determine whether the domain is discrete or continuous. Explain. (Section 3.2)

Question 9.

Answer:
The given table is:

From the above table,
The values of x have specific values of y
We know that,
The domain is defined as the set of all the values of x
Hence, from the above,
We can conclude that the domain of the given table is discrete

Question 10.

Answer:
The given table is:

From the above table
We can observe that the values of x have specific values of y
We know that,
The domain is defined as the set of all the values of x
Hence, from the above,
We can conclude that the domain of the given table is discrete

Question 11.
For w(x) = -2x + 7, find the value of x for which w(x) = -3. (Section 3.3)
Answer:
The given function is:
w (x) = -2x + 7 with w (x)  = -3
So,
-3 = -2x + 7
-2x = -3 – 7
-2x = -10
2x = 10
x = 10 / 5
x = 2
Hence, from the above,
We can conclude that the value of the given function is: 2

Graph the linear function. ( Section 3.3)

Question 12.
g(x) = x + 3
Answer:
The given function is:
g (x) = x + 3
Now,
put the values -2, -1, 0, 1, 2 in the place of x and find the values of f(x) toplot a graph [ Remember you can take any value and any number of values]
So,
The completed table for the given function is:

Hence,
The representation of the given function in the coordinate plane is:

Question 13.
p(x) = -3x – 1
Answer:
The given function is:
p (x) = -3x – 1
Now,
put the values -2, -1, 0, 1, 2 in the place of x and find the values of f(x) toplot a graph [ Remember you can take any value and any number of values]
So,
The completed table for the given function is:

Hence,
The representation of the given function in the coordinate plane is:

Question 14.
m(x) = \(\frac{2}{3}\)x
Answer:
The given function is:
m (x) = \(\frac{2}{3}\)x
Now,
put the values -2, -1, 0, 1, 2 in the place of x and find the values of f(x) to plot a graph [ Remember you can take any value and any number of values]
So,
The completed table for the given function is:

Hence,
The representation of the given function in the coordinate plane is:

Question 15.
The function m = 30 – 3r represents the amount m (in dollars) of money you have after renting r video games. (Section 3.1 and Section 3.2)
a. Identify the independent and dependent variables.
Answer:
The given function is:
m = 30 – 3r
We know that,
The independent variables are the values of x
The dependent variables are the values of y
Now,
Compare the given function with
y = mx + c
Hence,
The independent variable of the given function is: r
The dependent variable of the given function is: m

b. Find the domain and range of the function. Is the domain discrete or continuous? Explain.
Answer:
The given function is:
m = 30 – 3r
Now,
put the values -2, -1, 0, 1, 2 in the place of x and find the values of f(x) to plot a graph [ Remember you can take any value and any number of values]
So,
The completed table of the given function is:

We know that,
The domain is defined as the set of all the values of x
the range is defined as the set of all the values of y
Hence,
The domain of the given function is: {-2, -1, 0, 1, 2}
The range of the given function is: {36, 33, 30, 27, 24, 21}

c. Graph the function using its domain.
Answer:
The completed table for the given function from part (b) is:

Hence,
The representation of the given function using its domain in the coordinate plane is:

Question 16.
The function d(x) = 1375 – 110x represents the distance (in miles) a high-speed train is from its destination after x hours. (Section 3.3)
a. How far is the train from its destination after 8 hours?
Answer:
The given function is:
d (x) = 1375 – 110x
Where,
d (x) represents the distance (in miles)
x represents the time
Now,
It is given that we have to find the distance traveled by train after 8 hours i.e., the value of x is given
So,
d (x) = 1375 – 110 (8)
= 1375 – 880
= 495 miles
Hence, from the above,
We can conclude that the distance traveled by train after 8 hours is: 495 miles

b. How long does the train travel before reaching its destination?
Answer:
The given function is:
d (x) = 1375 – 110x
From part (a),
We find the value of d (x) as: 495 miles
So,
495 = 1375 – 110x
110x = 1375 – 495
110x = 880
x = 880 /110
x = 8 hours
Hence, from the above,
We can conclude that train travel for 8 hours before reaching its destination

Lesson 3.4 Graphing Linear Equations in Standard Form

Essential Question
How can you describe the graph of the equation Ax + By = C?
Answer:
When A and B are not both zero,
The graph of Ax + By = C is always a line.
Now,
Divide both sides by B
Because the form Ax + By = C can describe any line,
It is called the standard form of an equation for a line.

EXPLORATION 1
Using a Table to Plot Points
Work with a partner. You sold a total of $16 worth of tickets to a fundraiser. You lost track of how many of each type of ticket you sold. Adult tickets are $4 each. Child tickets are $2 each.

a. Let x represent the number of adult tickets. Let y represent the number of child tickets. Use the verbal model to write an equation that relates to x and y.
Answer:
It is given that you sold a total of $16 worth of tickets to a fundraiser. You lost track of how many of each type of ticket you sold. Adult tickets are $4 each. Child tickets are $2 each.
It is also given that
Let x be the number of adult tickets
Let y be the number of child tickets
So,
The total cost of tickets = (The cost of the child tickets + The cost of the adult tickets)
The total cost of tickets = (The number of children) × ( The cost of each child ticket ) + ( The numebr of adults ) × ( The cost of each adult ticket )
So,
16 = 2x + 4y
2 (x + 2y) = 16
x + 2y = 16 / 2
So,
x + 2y = 8
Hence, from the above,
We can conclude that the equation that relates x and y is:
x + 2y = 8

b. Copy and complete the table to show the different combinations of tickets you might have sold.
Answer:
The equation that represents the number of different tickets sold is:
x + 2y = 8
2y = 8 – x
y = \(\frac{8 – x}{2}\)
So,
The completed table that shows the different combinations of tickets you might have sold is:

c. Plot the points from the table. Describe the pattern formed by the points.
Answer:
The completed table that shows the different combinations of tickets you might have sold is from part (b) is:

The representation of the ordered pairs from the above table is:
(1, 4), (2, 3), (3, 3), (4, 2), (5, 2)
Hence,
The representation of the points from the table in the coordinate plane is:

Hence, from the graph,
We can observe that the pattern drawn by the points is a straight line

d. If you remember how many adult tickets you sold, can you determine how many child tickets you sold? Explain your reasoning.

Answer:
The given function from part (a) is in the form of
y = mx + c
Where,
y is the number of adult tickets
x is the number of child tickets
So,
If we know the number of adult tickets you sold, i.e., the value of y, then we can find the number of child tickets, i.e., the value of x by putting the value of y in the equation
y = mx + c
Hence, from the above,
We can conclude that if we know the number of adult tickets you sold, then we can find the number of child tickets you sold

EXPLORATION 2
Rewriting and Graphing an Equation
Work with a partner. You sold a total of $48 worth of cheese. You forgot how many pounds of each type of cheese you sold. Swiss cheese costs $8 per pound. Cheddar cheese costs $6 per pound.

a. Let x represent the number of pounds of Swiss cheese. Let y represent the number of pounds of cheddar cheese. Use the verbal model to write an equation that relates to x and y.
Answer:
it is given that you sold a total of $48 worth of cheese. You forgot how many pounds of each type of cheese you sold. Swiss cheese costs $8 per pound. Cheddar cheese costs $6 per pound.
It is also given that,
x represents the number of pounds of Swiss cheese
y represents the number of pounds of Cheddar cheese
Now,
The total cost = ( The cost of Swiss Cheese ) + ( The cost of Cheddar Cheese )
The total cost = ( The number of pounds of Swiss cheese ) × ( The cost of Swiss cheese ) + ( The number of pounds of Cheddar cheese ) × ( The cost of Cheddar cheese )
48 = 8x + 6y
8x + 6y = 48
2 (4x + 3y) = 48
4x + 3y = 48 / 4
4x + 3y = 12
Hence, from the above,
We can conclude that the equation that relates both x and y is:
4x + 3y = 12

b. Solve the equation for y. Then use a graphing calculator to graph the equation. Given the real-life context of the problem, find the domain and range of the function.
Answer:
The equation that relates to x and y from part (a) is:
4x + 3y = 12
3y = 12 – 4x
y = \(\frac{12 – 4x}{3}\)
So,
The representation of the equation in the coordinate plane is:

We know that,
The domain is the set of all the values of x in the graph
The range is the set of all the values of y in the graph
Now,
From the graph,
The domain of the equation is: { -4, -3, -2, -1, 0, 1, 2, 3 }
The range of the given equation is: {-1, -2, -3, -4, -5, -6, -7, -8, -9, 5, 6, 7, 8, 9, 10}

c. The x-intercept of a graph is the x-coordinate of a point where the graph crosses the x-axis. The y-intercept of a graph is the y-coordinate of a point where the graph crosses the y-axis. Use the graph to determine the x- and y-intercepts.
Answer:
The given equation is:
y = \(\frac{12 – 4x}{3}\)
It is given that the x-intercept of a graph is the x-coordinate of a point where the graph crosses the x-axis. The y-intercept of a graph is the y-coordinate of a point where the graph crosses the y-axis.
Now,
The graph from part (b) is:

Hence,
By observing the graph,
We can conclude that
The x-intercept is: 3
The y-intercept is: 4

d. How could you use the equation you found in part (a) to determine the x- and y-intercepts? Explain your reasoning.
Answer:
The given equation is:
y = \(\frac{12 – 4x}{3}\)
We know that,
We can obtain the x-intercept by making y term zero
We can obtain the y-intercept making x term zero
So,
The x-intercept is:
0 = \(\frac{12 – 4x}{3}\)
12 – 4x = 0
4x = 12
x = 12 / 4
x = 3
Hence,
The x-intercept is: 3
The y-intercept is:
y = \(\frac{12 – 4(0)}{3}\)
y = 12 / 3
y = 4
Hence,
The y-intercept is: 4

e. Explain the meaning of the intercepts in the context of the problem.
Answer:
An intercept of any function is a point where the graph of the function crosses, or intercepts, the x-axis or y-axis.  When the linear function is used to represent a real-world situation, the intercepts have significant meaning in the context of the problem.

Communicate Your Answer

Question 3.
How can you describe the graph of the equation Ax + By = C?
Answer:
When A and B are not both zero,
The graph of Ax + By = C is always a line.
Now,
Divide both sides by B
Because the form Ax + By = C can describe any line,
It is called the standard form of an equation for a line.

Question 4.
Write a real-life problem that is similar to those shown in Explorations 1 and 2.
Answer:
The real-life problem that is similar to those in Explorations  1 and 2 is:
The age of A is twice the age of B and the sum of the twice the age of A and the age of B is 24

3.4 Lesson

Monitoring Progress

Question 1.
y = -2.5
Answer:
The given equation is:
y = -2.5
Hence,
The representation of the given equation in the coordinate plane is:

Question 2.
x = 5
Answer:
The given equation is:
x = 5
Hence,
The representation of the given equation in the coordinate plane is:

Use intercepts to graph the linear equation. Label the points corresponding to the intercepts.

Question 3.
2x – y = 4
Answer:
The given equation is:
2x – y = 4
Rewrite the given equation in the standard form
We know that,
The standard form of the linear equation is:
y = mx + c
So,
y = 2x – 4
To find the value of x-intercept, put y = 0
2x – 4 = 0
2x = 4
x = 4 / 2
x = 2
To find the value of y-intercept, put x = 0
y = 2(0) – 4
y = -4
Hence,
The representation of the given equation corresponding to the intercepts in the coordinate plane is:

Question 4.
x + 3y = -9
Answer:
The given equation is:
x + 3y = -9
Rewrite the given equation in the standard form
We know that,
The standard form of the linear equation is:
y = mx + c
So,
3y = -9 – x
y = (-9 -x ) / 3
Now,
To find the value of x-intercept, put y = 0
-9 – x = 0
x = -9
To find the value of y-intercept, put x = 0
y = (-9 – 0) / 3
t = -9 / 3
y = -3
Hence,
The representation of the given equation corresponding to the intercepts in the coordinate plane is:

Question 5.
WHAT IF? You decide to rent tables from a different company. The situation can be modeled by the equation 4x + 6y = 180, where x is the number of small tables and y is the number of large tables. Graph the equation and interpret the intercepts.
Answer:
It is given that you decide to rent tables from a different company.
The given equation corresponding to the above situation is given as:
4x + 6y = 180
Where,
x is the number of small tables
y is the number of large tables
Now,
Rewrite the given equation in the standard form of the linear equation
We know that,
The standard form of the linear equation is:
y = mx + c
So,
6y = 180 – 4x
y = (180 – 4x) / 6
Now,
To find the value of the x-intercept, put y = 0
180 – 4x = 0
4x = 180
x = 180 / 4
x = 45
To find the value of the y-intercept, put x = 0
y = (180 – 4(0) ) / 6
y = 180 / 6
y = 30
Hence,
The representation of the given equation corresponding to the intercepts in the coordinate plane is:

Graphing Linear Equations in Standard Form 3.4 Exercises

Vocabulary and Core Concept Check

Question 1.
WRITING
How are x-intercepts and y-intercepts alike? How are they different?
Answer:

Question 2.
WHICH ONE did DOESN'T BELONG?
Which point does not belong with the other three? Explain your reasoning.

Answer:
The given points are:
a. (0, -3)
b. (0, 0)
c. (4, -3)
d. (4, 0)
From the above points,
We can observe that
(0, -3) belongs to the y-axis
(0, 0) belongs to the origin
(4, -3) does not belong either to the x-axis or y-axis
(4, 0) belongs to the x-axis
Hence, from the above,
We can conclude that (4, -3) does not belong with the other three

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, graph the linear equation.

Question 3.
x = 4
Answer:

Question 4.
y = 2
Answer:
The given equation is:
y = 2
Hence,
The representation of the given equation in the coordinate plane is:

Question 5.
y = -3
Answer:

Question 6.
x = -1
Answer:
The given equation is:
x = -1
Hence,
The representation of the given equation in the coordinate plane is:

In Exercises 7–12, find the x- and y-intercepts of the graph of the linear equation.

Question 7.
2x + 3y = 12
Answer:

Question 8.
3x + 6y = 24
Answer:
The given equation is:
3x + 6y = 24
Now,
To find the value of the x-intercept, put y = 0
So,
3x + 0 = 24
3x = 24
x = 24 / 3
x = 8
To find the value of the y-intercept, put x = 0
So,
0 + 6y = 24
6y = 24
y = 24 / 6
y = 4
Hence, from the above,
We can conclude that
The value of the x-intercept is: 8
The value of the y-intercept is: 4

Question 9.
-4x + 8y = -16
Answer:

Question 10.
-6x + 9y = -18
Answer:
The given equation is:
-6x + 9y = -18
Now,
To find the x-intercept, put y = 0
So,
-6x + 0 = -18
-6x = -18
6x = 18
x = 18 / 6
x = 3
To find the y-intercept, put x = 0
So,
0 + 9y = -18
9y = -18
y = -18 / 9
y = -2
Hence, from the above,
We can conclude that
The value of the x-intercept is: 3
The value of the y-intercept is: -2

Question 11.
3x – 6y = 2
Answer:

Question 12.
-x + 8y = 4
Answer:
The given equation is:
-x + 8y = 4
Now,
To find the x-intercept, put y = 0
So,
-x + 0 = 4
-x = 4
x = -4
To find the y-intercept, put x = 0
So,
0 + 8y = 4
8y = 4
y = 4 / 8
y = 1 / 2
Hence, from the above,
We can conclude that
The value of the x-intercept is: -4
The value of the y-intercept is: 1/2

In Exercises 13–22, use intercepts to graph the linear equation. Label the points corresponding to the intercepts.

Question 13.
5x + 3y = 30
Answer:

Question 14.
4x + 6y = 12
Answer:
The given equation is:
4x + 6y = 12
Now,
To find the x-intercept, put y = 0
So,
4x + 0 = 12
4x = 12
x = 12 / 4
x = 3
To find the y-intercept, put x = 0
0 + 6y = 12
6y = 12
y = 12 / 6
y = 2
Hence,
The representation of the given equation along with the intercepts in the coordinate plane is:

Question 15.
-12x + 3y = 24
Answer:

Question 16.
-2x + 6y = 18
Answer:
The given equation is:
-2x + 6y = 18
Now,
To find the x-intercept, put y = 0
So,
-2x + 0 = 18
-2x = 18
x = -18 / 2
x = -9
To find the y-intercept, put x = 0
So,
0 + 6y = 18
6y = 18
y = 18 / 6
y = 3
Hence,
The representation of the given equation along with the intercepts in the coordinate plane is:

Question 17.
-4x + 3y = -30
Answer:

Question 18.
-2x + 7y = -21
Answer:
The given equation is:
-2x + 7y = -21
Now,
To find the value of the x-intercept, put y = 0
-2x + 0 = -21
-2x = -21
2x = 21
x = 21/2
To find the value of the y-intercept, put x = 0
So,
0 + 7y = -21
7y = -21
y = -21 / 7
y = -3
Hence,
The representation of the given equation along with the intercepts in the coordinate plane is:

Question 19.
-x + 2y = 7
Answer:

Question 20.
3x – y = -5
Answer:
The given equation is:
3x – y = -5
Now,
To find the x-intercept, put y = 0
So,
3x – 0 = -5
3x =  -5
x = -5/3
To find the y-intercept, put x = 0
So,
0 – y = -5
-y = -5
y = 5
Hence,
The representation of the given equation along with the intercepts in the coordinate plane is:

Question 21.
–\(\frac{5}{2}\)x + y = 10
Answer:

Question 22.
–\(\frac{1}{2}\)x + y = -4
Answer:
The given equation is:
–\(\frac{1}{2}\)x + y = -4
Now,
To find the x-intercept, put y =0
So,
–\(\frac{1}{2}\)x + 0 = -4
–\(\frac{1}{2}\)x = -4
-x = -4(2)
-x = -8
x = 8
To find the y-intercept, put x = 0
So,
0 + y = -4
y = -4
Hence,
The representation of the given equation along with the intercepts in the coordinate plane is:

Question 23.
MODELING WITH MATHEMATICS
A football team has an away game, and the bus breaks down. The coaches decide to drive the players to the game in cars and vans. Four players can ride in each car. Six players can ride in each van. There are 48 players on the team. The equation 4x + 6y = 48 models this situation, where x is the number of cars and y is the number of vans.
a. Graph the equation. Interpret the intercepts.
b. Find four possible solutions in the context of the problem.
Answer:

Question 24.
MODELING WITH MATHEMATICS
You are ordering shirts for the math club at your school. Short-sleeved shirts cost $10 each. Long-sleeved shirts cost $12 each. You have a budget of $300 for the shirts. The equation 10x + 12y = 300 models the total cost, where x is the number of short-sleeved shirts and y is the number of long-sleeved shirts.
a. Graph the equation. Interpret the intercepts.
Answer:
It is given that you are ordering shirts for the math club at your school. Short-sleeved shirts cost $10 each. Long-sleeved shirts cost $12 each. You have a budget of $300 for the shirts.
The above situation is represented by the equation
10x + 12y = 300
where,
x is the number of short-sleeved shirts
y is the number of long-sleeved shirts.
Now,
The given equation is:
10x + 12y = 300
To find the x-intercept, put y =0
So,
10x + 12 (0) = 300
10x = 300
x = 300 / 10
x = 30
To find the y-intercept, put x = 0
So,
10 (0) + 12y = 300
12y = 300
y = 300 / 12
y = 25
Hence,
The representation of the given equation along with the intercepts in the coordinate plane is:

b. Twelve students decide they want short-sleeved shirts. How many long-sleeved shirts can you order?
Answer:
The given equation is:
10x + 12y = 300
where,
x is the number of short-sleeved shirts
y is the number of long-sleeved shirts.
It is given that 12 students decided they want short-sleeved shirts
So,
The number of short-sleeved shirts = 12
So,
10 (12) + 12y = 300
120 + 12y = 300
12y = 300 – 120
12y = 180
y = 180 / 12
y = 15
Hence, from the above,
We can conclude that the number of long-sleeved shirts is: 15

ERROR ANALYSIS
In Exercises 25 and 26, describe and correct the error in finding the intercepts of the graph of the equation.

Question 25.

Answer:

Question 26.

Answer:
The given equation is:
4x + 10y = 20
Now,
To find the value of the x-intercept, put y = 0
So,
4x + 10 (0) = 20
4x = 20
x = 20 / 4
x = 5
To find the value of the y-intercept, put x = 0
So,
4 (0) + 10y = 20
10y = 20
y = 20 / 10
y = 2
Hence, from the above,
We can conclude that
The x-intercept is: (5, 0)
The y-intercept is: (0, 2)

Question 27.
MAKING AN ARGUMENT
You overhear your friend explaining how to find intercepts to a classmate. Your friend says, "When you want to find the x-intercept, just substitute 0 for x and continue to solve the equation." Is your friend's explanation correct? Explain.
Answer:

Question 28.
ANALYZING RELATIONSHIPS
You lose track of how many 2-point baskets and 3-point baskets a team makes in a basketball game. The team misses all the 1-point baskets and still scores 54 points. The equation 2x + 3y = 54 models the total points scored, where x is the number of 2-point baskets made and y is the number of 3-point baskets made.
a. Find and interpret the intercepts.
Answer:
It is given that you lose track of how many 2-point baskets and 3-point baskets a team makes in a basketball game.
It is also given that the team misses all the 1-point baskets and still scores 54 points.
The above situation can be described by the equation:
2x + 3y = 54
where,
x is the number of 2-point baskets made
y is the number of 3-point baskets made.
So,
To find the value of the x-intercept, put y = 0
So,
2x = 54
x = 54 / 2
x = 27
To find the value of the y-intercept, put x = 0
3y = 54
y = 54 / 3
y = 18
Hence, from the above,
We can conclude that
The value of the x-intercept is: 27
The value of the y-intercept is: 18

b. Can the number of 3-point baskets made be odd? Explain your reasoning.
Answer:
From part (a),
The number of 3-point baskets can be represented as: y
So,
To make the 3-point baskets odd, the multiples of 3 must be odd
But, in this scenario, it is not possible
Hence, from the above,
We can conclude that the number of 3-point baskets made not be odd

c. Graph the equation. Find two more possible solutions in the context of the problem.
Answer:

The given equation is:
2x + 3y = 54
From the graph,
We can observe that there are many solutions like (3, 16), (6, 14), (10, 11) and so on
Now,
To satisfy the above equation, substitute the points we obtained from the graph
Hence, from the above,
We can conclude that the solutions to the given equation are: (3, 16) and (6, 14)

MULTIPLE REPRESENTATIONS
In Exercises 29–32, match the equation with its graph.

Question 29.
5x + 3y = 30
Answer:

Question 30.
5x + 3y = -30
Answer:
The given equation is:
5x + 3y = -30
Now,
To find the value of the x-intercept, put y = 0
So,
5x = -30
x = -30 / 5
x = -6
To find the value of the y-intercept, put x = 0
So,
3y = -30
y = -30 / 3
y = -10
Hence, from the above,
We can conclude that the graph C) matches the given equation

Question 31.
5x – 3y = 30
Answer:

Question 32.
5x – 3y = -30
Answer:
The given equation is:
5x – 3y = -30
Now,
To find the value of the x-intercept, put y = 0
So,
5x = -30
x = -30 / 5
x = -6
To find the value of the y-intercept, put x = 0
So,
-3y = -30
3y = 30
y = 30 / 3
y = 10
Hence, from the above,
We can conclude that the graph B) matches the given equation

Question 33.
MATHEMATICAL CONNECTIONS
Graph the equations x = 5, x = 2, y = -2, and y = 1. What enclosed shape do the lines form? Explain your reasoning.
Answer:

Question 34.
HOW DO YOU SEE IT? You are organizing a class trip to an amusement park. The cost to enter the park is $30. The cost to enter with a meal plan is $45. You have a budget of $2700 for the trip. The equation 30x + 45y = 2700 models the total cost for the class to go on the trip, where x is the number of students who do not choose the meal plan and y is the number of students who do choose the meal plan.

a. Interpret the intercepts of the graph.
Answer:
The given graph is:

We know that,
The x-intercept is the point that cuts the x-axis and y must be 0
The y-intercept is the point that cuts the y-axis and x must be 0
Hence,
From the graph,
We can conclude that
The x-intercept is: (90, 0)
The y-intercept is: (0, 60)

b. Describe the domain and range in the context of the problem.
Answer:
The given graph is:

We know that,
The domain is defined as the set of all the values of x
The range is defined as the set of all the values of y
Hence,
The domain of the given function is: {0, 10, 20, 30, 40, 50, 60, 70, 80, 90}
The range of the given function is: {0, 10, 20, 30, 40, 50, 60}

Question 35.
REASONING
Use the values to fill in the equation so that the x-intercept of the graph is -10 and the y-intercept of the graph is 5.

Answer:

Question 36.
THOUGHT-PROVOKING
Write an equation in the standard form of a line whose intercepts are integers. Explain how you know the intercepts are integers.
Answer:
We know that,
The equation in the standard form is:
y = mx + c
Now,
To find the value of the x-intercept, put y = 0
So,
mx = -c
x = –\(\frac{c}{m}\)
To find the value of the y-intercept, put x = 0
So,
y = 0 + c
y = c
Now, from the above
We can observe that to make the x and y-intercepts integers, the values of the intercepts must be the multiples of the variable c

Question 37.
WRITING
Are the equations of horizontal and vertical lines written in standard form? Explain your reasoning.
Answer:

Question 38.
ABSTRACT REASONING
The x- and y-intercepts of the graph of the equation 3x + 5y = k are integers. Describe the values of k. Explain your reasoning.
Answer:
The given equation is:
3x + 5y = k
To find the value of the x-intercept, put y = 0
So,
3x = k
So,
The values of k so that k become integer is 0, 1, 2, and so on
To find the value of the y-intercept, put x = 0
So,
5y = k
So,
The values of k so that k becomes integer is 0, 1, 2, and so on
Hence, from the above,
We can conclude that the values of k so that the x and y-intercepts become integers are: 0, 1, 2, and so on

Maintaining Mathematical Proficiency

Simplify the expression.

Question 39.

Answer:

Question 40.

Answer:
The given expression is:
\(\frac{14 -18}{0 – 2}\)
= \(\frac{-4}{-2}\)
= \(\frac{4}{2}\)
= 2
Hence, from the above,
We can conclude that the value of the given expression is: 2

Question 41.

Answer:

Question 42.

Answer:
The given expression is:
\(\frac{12 – 17}{-5 – (-2)}\)
= \(\frac{12 – 17}{-5 + 2}\)
= \(\frac{-5}{-3}\)
= \(\frac{5}{3}\)
Hence, from the above,
We can conclude that the value of the given expression is: \(\frac{5}{3}\)

Lesson 3.5 Graphing Linear Equations in Slope-Intercept Form

EXPLORATION 1
Finding Slopes and y-Intercepts
Work with a partner. Find the slope and y-intercept of each line.

Answer:

We know that,
The standard form of a linear equation is:
y = mx + c
Where,
m is the slope
c is the y-intercept
Now,
a.
The given equation is:
y = \(\frac{2}{3}\)x + 2
Now,
Compare the given equation with the standard form of the linear equation
By comparison,
We get,
m = \(\frac{2}{3}\) and c = 2

b.
The given equation is:
y = -2x – 1
Now,
Compare the given equation with the standard form of the linear equation
By comparison,
We get,
m = -2 and c = -1

EXPLORATION 2
Writing a Conjecture
Work with a partner. Graph each equation. Then copy and complete the table. Use the completed table to write a conjecture about the relationship between the graph of y = mx + b and the values of m and b.


Answer:
The given table is:

Now,
The representation of the given equations in the coordinate plane is as follows:
Now,
a.
The given equation is:
y = –\(\frac{2}{3}\)x + 3
Hence,
The representation of the given equation in the coordinate plane is:

b.
The given equation is:
y = 2x – 2
Hence,
The representation of the given equation in the coordinate plane is:

c.
The given equation is:
y = -x + 1
Hence,
The representation of the given equation in the coordinate plane is:

d.
The given equation is:
y = x – 4
Hence,
The representation of the given equation in the coordinate plane is:

Now,
We know that,
The standard form of a linear equation is:
y = mx + b
Where,
m is the slope
b is the y-intercept
Hence,
The completed table for the above equations' slopes and y-intercepts is:

Communicate Your Answer

Question 3.
How can you describe the graph of the equation y = mx + b?
a. How does the value of m affect the graph of the equation?
b. How does the value of b affect the graph of the equation?
c. Check your answers to parts (a) and (b) by choosing one equation from Exploration 2 and (1) varying only m and (2) varying only b.
Answer:
The graph of the equation
y = mx + b
is a straight line
Now,
a.
The value of m affects the steepness of the line. It also describes the direction (positive or negative). The value of m defines the constant rate of change of variables

b.
The value of b affects the line to where it should have the point of intersection with the y-axis

c.
The examples of how m and b varies are as follows:
y =-3x – 3
y = 2x + 8
y = -3x + 6
Hence,
The slopes and y-intercepts of the above equations are:
m = -3 and b = -3
m = 2 and b = 8
m = -3 and b = 6

3.5 Lesson

Monitoring Progress

Describe the slope of the line. Then find the slope.

Question 1.

Answer:
The given graph is:

From the graph,
The given points are:
(-4, 3), (1, 1)
We know that,
The slope of the line when the two points are given is:
m = \(\frac{y2 – y1}{x2 – x1}\)
So,
The points are represented as (x, y)
So,
The first point is represented as (x1, y1)
The second point is represented as (x2, y2)
So,
(x1, y1) = (-4, 3) and (x2, y2) = (1, 1)
Hence,
m = \(\frac{1 – 3}{1 – [-4]}\)
m = \(\frac{-2}{1 + 4}\)
m = \(\frac{-2}{5}\)
Hence, from the above,
We can conclude that the slope of the given graph is: \(\frac{-2}{5}\)

Question 2.

Answer:
The given graph is:

From the graph,
The given points are:
(3, 3), (-3, 1)
We know that,
The slope of the line when the two points are given is:
m = \(\frac{y2 – y1}{x2 – x1}\)
So,
The points are represented as (x, y)
So,
The first point is represented as (x1, y1)
The second point is represented as (x2, y2)
So,
(x1, y1) = (3, 3) and (x2, y2) = (-3, 1)
Hence,
m = \(\frac{1 – 3}{-3 – 3}\)
m = \(\frac{-2}{6}\)
m = \(\frac{-1}{3}\)
Hence, from the above,
We can conclude that the slope of the given graph is: \(\frac{-1}{3}\)

Question 3.

Answer:
The given graph is:

From the graph,
The given points are:
(5, 4), (2, -3)
We know that,
The slope of the line when the two points are given is:
m = \(\frac{y2 – y1}{x2 – x1}\)
So,
The points are represented as (x, y)
So,
The first point is represented as (x1, y1)
The second point is represented as (x2, y2)
So,
(x1, y1) = (5, 4) and (x2, y2) = (2, -3)
Hence,
m = \(\frac{-3 – 4}{2 – 5}\)
m = \(\frac{-7}{-3}\)
m = \(\frac{7}{3}\)
Hence, from the above,
We can conclude that the slope of the given graph is: \(\frac{7}{3}\)

Monitoring Progress

The points represented by the table lie on a line. How can you find the slope of the line from the table? What is the slope of the line?

Question 4.

Answer:
The given table is:

From the above table,
The representations of the values of x and y in the form of ordered pairs are:
(2, 10), (4, 15), (6, 20), (8, 25)
Now,
To find the slope of a line,
Take any 2 ordered pairs and find the slope
Now,
Let,
(x1, y1) = (2, 10) and (x2, y2) = (4, 15)
We know that,
The slope of the line when the two points are given is:
m = \(\frac{y2 – y1}{x2 – x1}\)
Hence,
m = \(\frac{15 – 10}{4 – 2}\)
m = \(\frac{5}{2}\)
Hence, from the above,
We can conclude that the slope of the given graph is: \(\frac{5}{2}\)

Question 5.

Answer:
The given table is:

From the above table,
The representations of the values of x and y in the form of ordered pairs are:
(5, -12), (5, -9), (5, -6), (5, -3)
Now,
To find the slope of a line,
Take any 2 ordered pairs and find the slope
Now,
Let,
(x1, y1) = (5, -12) and (x2, y2) = (5, -9)
We know that,
The slope of the line when the two points are given is:
m = \(\frac{y2 – y1}{x2 – x1}\)
Hence,
m = \(\frac{-9 – [-12]}{5 – 5}\)
m = \(\frac{3}{0}\)
Hence, from the above,
We can conclude that the slope of the given graph is: undefined

Find the slope and the y-intercept of the graph of the linear equation.

Question 6.
y = -6x + 1
Answer:
The given equation is:
y = -6x + 1
We know that,
The standard representation of a linear equation is:
y  mx + c
Where,
m is the slope
c is the y-intercept
Now,
Compare the given equation with the standard representation of the linear equation
Hence,
The values of m and c are: -6 and 1

Question 7.
y = 8
Answer:
The given equation is:
y = 8
Rewrite the given equation as:
y = 0x + 8
We know that,
The standard representation of a linear equation is:
y  mx + c
Where,
m is the slope
c is the y-intercept
Now,
Compare the given equation with the standard representation of the linear equation
Hence,
The values of m and c are: 0 and 8

Question 8.
x + 4y = -10
Answer:
The given equation is:
x + 4y = -10
Rewrite the given equation as:
4y = -10 – x
y = \(\frac{-10 – x}{4}\)
y = \(\frac{-x}{4}\) – \(\frac{10}{4}\)
We know that,
The standard representation of a linear equation is:
y  mx + c
Where,
m is the slope
c is the y-intercept
Now,
Compare the given equation with the standard representation of the linear equation
Hence,
The values of m and 'c' are: –\(\frac{1}{4}\) and –\(\frac{10}{4}\)

Graph the linear equation. Identify the x-intercept.

Question 9.
y = 4x – 4
Answer:
The given equation is:
y = 4x – 4
Now,
To find the x-intercept, put y = 0
So,
0 = 4x – 4
4x = 4
x = 4 / 4
x = 1
Hence,
The representation of the given equation along with the x-intercept in the coordinate plane is:

Question 10.
3x + y = -3
Answer:
The given equation is:
3x + y = -3
Now,
To find the x-intercept, put y = 0
So,
3x + 0 = -3
3x = -3
x = -3 / 3
x = -1
Hence,
The representation of the given equation along with the x-intercept in the coordinate plane is:

Question 11.
x + 2y = 6
Answer:
The given equation is:
x + 2y = 6
Now,
To find the x-intercept, put y = 0
So,
x + 0 = 6
x = 6
Hence,
The representation of the given equation along with the x-intercept in the coordinate plane is:

Question 12.
A linear function h models a relationship in which the dependent variable decreases 2 units for every 5 units the independent variable increases. Graph h when h(0) = 4. Identify the slope, y-intercept, and x-intercept of the graph.
Answer:
It is given that a linear function h models a relationship in which the dependent variable decreases 2 units for every 5 units the independent variable increases.
We know that,
The independent variable is: x
the dependent variable is: y
So,
So,
x = -2 and y = +5
So,
We can say that the slope is represented as:
m = \(\frac{-2}{5}\)
It is also given that
h (0) = 2
So,
y = 2 at x = 0
Hence,
The y-intercept is: (0, 2)
Hence,
The representation of the equation in the standard form is:
y = mx + c
y = –\(\frac{2}{5}\)x + 2
Now,
The value of the x-intercept can be obtained by putting y = 0
So,
0 = –\(\frac{2}{5}\)x + 2
–\(\frac{2}{5}\)x = -2
\(\frac{2}{5}\)x = 2
x = \(\frac{5 × 2}{2}\)
x = 5
Hence, from the above,
We can conclude that
The slope is: –\(\frac{2}{5}\)
The x-intercept is: (5, 0)
The y-intercept is: (0, 2)

Question 13.
WHAT IF? The elevation of the submersible is modeled by h(t) = 500t – 10,000.
(a) Graph the function and identify its domain and range.
Answer:
The given function is:
h(t) = 500t – 10,000
So,
The representation of the given function in the coordinate plane is:

From the above graph,
We can say that the given equation is parallel to the y-axis
Hence,
There are no values for the domain since there are no values of x
The range of the given function is: -20,000 ≤ t ≤ 20,000

(b) Interpret the slope and the intercepts of the graph.
Answer:
The given function is:
h (t) = 500t – 10,000
Compare the given equation with the standard linear equation y = mx + c
So,
m = 500
c = -10,000
Now,
To find the x-intercept, put y = 0 or h (t) = 0
So,
500t – 10,000 = 0
500t = 10,000
t = 10,000 / 500
t = 20 or x = 20
To find the y-intercept, put x = 0 or t = 0
So,
h (t) = 500 (0) – 10,000
h (t) = -10,000
Hence, from the above,
We can conclude that
The slope is: 500
The x-intercept is: 20
The y-intercept is: -10,000

Graphing Linear Equations in Slope-Intercept Form 3.5 Exercises

Vocabulary and Core Concept

Question 1.
COMPLETE THE SENTENCE
The ________ of a nonvertical line passing through two points is the ratio of the rise to the run.
Answer:

Question 2.
VOCABULARY
What is a constant function? What is the slope of a constant function?
Answer:
A "Constant function" is a function where the output (y) is the same for every input (x) value.
The slope for a constant function will be 0

Question 3.
WRITING
What is the slope-intercept form of a linear equation? Explain why this form is called the slope-intercept form.
Answer:

Question 4.
WHICH ONE did DOESN'T BELONG? Which equation does not belong with the other three? Explain your reasoning.

Answer:
The given equations are:
a. y = -5x – 1
b. 2x – y = 8
c. y = x + 4
d. y = -3x + 13
Now,
Rewrite all the above equations in the form of
y = mx + c
So,
Now,
a.
The given equation is:
y = -5x – 1
Compare the above equation with
y = mx + c
So,
We will get,
m = -5 and c = -1

b.
The given equation is:
2x – y = 8
y = 2x – 8
Compare the above equation with
y = mx + c
So,
We will get
m = 8 and c = -8

c.
The given equation is:
y = x + 4
Compare the above equation with
y = mx + c
So,
We will get
m = 1 and c = 4

d.
The given equation is:
y = -3x + 13
Compare the above equation with
y = mx + c
So,
We will get
m = -3 and c = 3
Hence, from the above,
We can conclude that equation (b) does not belong with the other three since the slope of equation (b) is even whereas all the other slops are odd

Monitoring Progress and Modeling with Mathematics

In Exercises 5–8, describe the slope of the line. Then find the slope.

Question 5.

Answer:

Question 6.

Answer:
The given graph is:

From the graph,
We can observe that the slope falls from right to left
So,
The slope is positive
Now,
We have to represent the first pair as (x1, y1) and the second pair as (x2, y2)
Now,
The representation of the values of x and y in the form of ordered pairs to find a slope is:
(4, 3), (1, -1)
We know that,
m = \(\frac{y2 – y1}{x2 – x1}\)
m = \(\frac{-1 – 3}{1 – 4}\)
m = \(\frac{-4}{-3}\)
m = \(\frac{4}{3}\)
Hence, from the above,
We can conclude that the slope of the line is: \(\frac{4}{3}\)

Question 7.

Answer:

Question 8.

Answer:
The given graph is:

From the graph,
We can observe that the slope falls from left to right
So,
The slope is negative
Now,
We have to represent the first pair as (x1, y1) and the second pair as (x2, y2)
Now,
The representation of the values of x and y in the form of ordered pairs to find a slope is:
(0, 3), (5, -1)
We know that,
m = \(\frac{y2 – y1}{x2 – x1}\)
m = \(\frac{-1 – 3}{5 – 0}\)
m = \(\frac{-4}{5}\)
m = –\(\frac{4}{5}\)
Hence, from the above,
We can conclude that the slope of the line is: –\(\frac{4}{5}\)

In Exercises 9–12, the points represented by the table lie on a line. Find the slope of the line.

Question 9.

Answer:

Question 10.

Answer:
The given table is:

We know that,
To find the slope from a given table, we can take any x and y pair from the table
We have to represent the first pair as (x1, y1) and the second pair as (x2, y2)
Now,
The representation of the values of x and y in the form of ordered pairs to find a slope is:
(-1, -6), (2, -6)
We know that,
m = \(\frac{y2 – y1}{x2 – x1}\)
m = \(\frac{-6 + 6}{2 + 1}\)
m = \(\frac{0}{3}\)
m = 0
Hence, from the above,
We can conclude that the slope of the line is: 0

Question 11.

Answer:

Question 12.

Answer:
The given table is:

We know that,
To find the slope from a given table, we can take any x and y pair from the table
We have to represent the first pair as (x1, y1) and the second pair as (x2, y2)
Now,
The representation of the values of x and y in the form of ordered pairs to find a slope is:
(-4, 2), (-3, -5)
We know that,
m = \(\frac{y2 – y1}{x2 – x1}\)
m = \(\frac{-5 – 2}{-3 – [-4]}\)
m = \(\frac{-7}{1}\)
m = -7
Hence, from the above,
We can conclude that the slope of the line is: -7

Question 13.
ANALYZING A GRAPH
The graph shows the distance y(in miles) that a bus travels in x hours. Find and interpret the slope of the line.

Answer:

Question 14.
ANALYZING A TABLE
The table shows the amount x(in hours) of time you spend at a theme park and the admission fee y (in dollars) to the park. The points represented by the table lie on a line. Find and interpret the slope of the line.

Answer:
The given table is:

It is given that the table shows the amount x(in hours) of time you spend at a theme park and the admission fee y (in dollars) to the park and the points represented by the table lie on a line.
Now,
The representations of the x and y values in the form of ordered pairs i.e., (x, y)
So,
Let (6, 54.99) and (7, 54.99) be the 2 ordered pairs
We know that,
m = \(\frac{y2 – y1}{x2 – x1}\)
m = \(\frac{54.99 – 54.99}{7 – 6}\)
m = \(\frac{0}{1}\)
m = 0
Hence, from the above,
We can conclude that the slope of the line is: 0

In Exercises 15–22, find the slope and the y-intercept of the graph of the linear equation.

Question 15.
y = -3x + 2
Answer:

Question 16.
y = 4x – 7
Answer:
The given equation is:
y = 4x – 7
Compare the given equation with the standard form of the linear equation
y = mx + c
By comparing, we will get
m = 4 and c = -7
Hence, from the above,
We can conclude that
The slope of the given equation (m) is: 4
The y-intercept of the given equation is: -7

Question 17.
y = 6x
Answer:

Question 18.
y = -1
Answer:
The given equation is:
y = -1
Rewrite the given equation in the form of
y = mx + c
So,
y = 0x – 1
Compare the given equation with the standard form of the linear equation
y = mx + c
By comparing, we will get
m = 0 and c = -1
Hence, from the above,
We can conclude that
The slope of the given equation (m) is: 0
The y-intercept of the given equation is: -1

Question 19.
-2x + y = 4
Answer:

Question 20.
x + y = -6
Answer:
The given equation is:
x + y = -6
Rewrite the given equation in the form of the linear equation
y = mx + c
So,
y = -x – 6
Compare the given equation with the standard form of the linear equation
y = mx + c
By comparing, we will get
m = -1 and c = -6
Hence, from the above,
We can conclude that
The slope of the given equation (m) is: -1
The y-intercept of the given equation is: -6

Question 21.
-5x = 8 – y
Answer:

Question 22.
0 = 1 – 2y + 14x
Answer:
The given equation is:
0 =1 – 2y + 14x
Rewrite the give equation in the form of
y = mx + c
So,
2y = 14x + 1
Divide by 2 into both sides
y = (14 / 2)x + (1 / 2)
y = 7x + \(\frac{1}{2}\)
Compare the given equation with the standard form of the linear equation
y = mx + c
By comparing, we will get
m = 7 and c = \(\frac{1}{2}\)
Hence, from the above,
We can conclude that
The slope of the given equation (m) is: 7
The y-intercept of the given equation is:\(\frac{1}{2}\)

ERROR ANALYSIS
In Exercises 23 and 24, describe and correct the error in finding the slope and the y-intercept of the graph of the equation.

Question 23.

Answer:

Question 24.

Answer:
The given equation is:
y = 3x – 6
Compare the above equation with the standard form of the linear equation
y = mx + c
So,
We will get
m = 3 and c = -6
Hence, from the above,
We can conclude that
The slope of the given equation (m) is: 3
The y-intercept of the given equation is: -6

In Exercises 25–32, graph the linear equation. Identify the x-intercept.

Question 25.
y = -x + 7
Answer:

Question 26.
y = \(\frac{1}{2}\)x + 3
Answer:
The given linear equation is:
y = \(\frac{1}{2}\)x + 3
To find the x-intercepy, put y = 0
0 = \(\frac{1}{2}\)x + 3
–\(\frac{1}{2}\)x = 3
x = 3 (-2)
x = -6
Hence,
The representation of the given linear equation in the coordinate plane is:

Question 27.
y = 2x
Answer:

Question 28.
y = -x
Answer:
The given linear equation is:
y = -x
Rewrite the given equation in the form of y = mx + c
So,
y = -x + 0
To find the x-intercept, put y = 0
So,
0 = -x + 0
x = 0
Hence,
The representation of the given linear equation along with the x-intercept in the coordinate plane is:

Question 29.
3x + y = -1
Answer:

Question 30.
x + 4y = 8
Answer:
The given linear equation is:
x + 4y = 8
4y = -x + 8
Divide by 4 into both sides
y = –\(\frac{1}{4}\)x + (8 / 4)
y = –\(\frac{1}{4}\)x + 2
To find the x-intercept, put y = 0
0 = –\(\frac{1}{4}\)x + 2
\(\frac{1}{4}\)x = 2
x = 2(4)
x = 8
Hence,
The representation of the given linear equation along with the x-intercept in the coordinate plane is:

Question 31.
-y + 5x = 0
Answer:

Question 32.
2x – y + 6 = 0
Answer:
The given linear equation is:
2x – y + 6 = 0
So,
y = 2x + 6
To find the x-intercept, put y = 0
So,
0 = 2x + 6
-2x = 6
x = -6 / 2
x = -3
Hence,
The representation of the given linear equation along with the x-intercept in the coordinate plane is:

In Exercises 33 and 34, graph the function with the given description. Identify the slope, y-intercept, and x-intercept of the graph.

Question 33.
A linear function f models a relationship in which the dependent variable decreases 4 units for every 2 units the independent variable increases. The value of the function at 0 is -2.
Answer:

Question 34.
A linear function h models a relationship in which the dependent variable increases 1 unit for every 5 units the independent variable decreases. The value of the function at 0 is 3.
Answer:
It is given that a linear function h models a relationship in which the dependent variable increases 1 unit for every 5 units the independent variable decreases.
We know that,
The independent variable is: x
The dependent variable is: y
So,
Slope (m) = y / x
m = 1 / (-5)
m = –\(\frac{1}{5}\)
It is given that the value of the function at 0 is 3
So,
The value of y at x = 0 is: 3
Hence,
The y-intercept of the given function is: 3
Hence,
The representation of the slope and the y-intercept in the standard form of the linear function s:
y = mx + c
So,
y = –\(\frac{1}{5}\)x + 3
To find the x-intercept, put y = 0
So,
0 = –\(\frac{1}{5}\)x + 3
\(\frac{1}{5}\)x = 3
x = 3(5)
x = 15
To find the y-intercept, put x = 0
So,
y = 3
Hence, from the above,
We can conclude that
The slope of the given function is: –\(\frac{1}{5}\)
The x-intercept is: 15
The y-intercept is: 3
The representation of the given function in the coordinate plane is:

Question 35.
GRAPHING FROM A VERBAL DESCRIPTION
A linear function r models the growth of your right index fingernail. The length of the fingernail increases 0.7 millimeters every week. Graph r when r(0) = 12. Identify the slope and interpret the y-intercept of the graph.
Answer:

Question 36.
GRAPHING FROM A VERBAL DESCRIPTION
A linear function m models the amount of milk sold by a farm per month. The amount decreases 500 gallons for every $1 increase in price. Graph m when m(0) = 3000. Identify the slope and interpret the x- and y-intercepts of the graph.
Answer:
It is given that the amount of milk decreases 500 gallons for every $1 increase
From the above,
We can say that,
The price is the independent variable
So,
x represents the price
The amount of milk is the dependent variable
So,
y represents the amount of milk
So,
Slope = y / x
Slope = -500 / 1
So,
Slope (m) = -500
It is given that,
m (0) = 3000
We know that,
m (0) is the functional representation of y
So,
y = 3000 when x = 0
Hence,
The y-intercept is: 3000
Now,
The representation of the slope and the y-intercept in the standard form of the linear equation is:
y = ,x + c
y = -500x + 3000
Now,
To find the x-intercept, put y = 0
So,
0 = -500x + 3000
500x = 3000
x = 3000 / 500
x = 6
To find the y-intercept, put x = 0
So,
y = -500 (0) + 3000
y = 3000
Hence, from the above,
We can conclude that
The slope of the given equation is: -500
The c-intercept or y-intercept of the given equation is: 3000
The x-intercept of the given equation is: 6

Question 37.
MODELING WITH MATHEMATICS
The function shown models the depth d (in inches) of snow on the ground during the first 9 hours of a snowstorm, where t is the time (in hours) after the snowstorm begins.

a. Graph the function and identify its domain and range.
b. Interpret the slope and the d-intercept of the graph.
Answer:

Question 38.
MODELING WITH MATHEMATICS
The function c(x) = 0.5x + 70 represents the cost c (in dollars) of renting a truck from a moving company, where x is the number of miles you drive the truck.
a. Graph the function and identify its domain and range.
Answer:
The given function is:
c (x) = 0.5x + 70
Where,
represents the cost c (in dollars) of renting a truck from a moving company,
x is the number of miles you drive the truck.
Hence,
The representation of the given function in the coordinate plane is:

We know that,
The domain is the set of all the values of x that holds the given equation true
The range is defined as the set of all the values of y that holds the given equation true
Hence,
The domain of the given function is: -100 ≤ x ≤ 60
The range of the given function is: 20 ≤y ≤ 70

b. Interpret the slope and the c-intercept of the graph.
Answer:
The given function is:
c (x) = 0.5x + 70
Compare the above equation with the standard form of the linear function
y = mx + c
SO,
By comparing, we get
m = 0.5 and c = 70
Hence, from the above,
We can conclude that
The slope of the given function is: 0.5
The c-intercept is: 70

Question 39.
COMPARING FUNCTIONS
A linear function models the cost of renting a truck from a moving company. The table shows the cost y (in dollars) when you drive the truck x miles. Graph the function and compare the slope and the y-intercept of the graph with the slope and the c-intercept of the graph in Exercise 38.

Answer:

ERROR ANALYSIS
In Exercises 40 and 41, describe and correct the error in graphing the function.

Question 40.

Answer:
The given equation is:
y + 1 = 3x
Rewrite the above equation in the form of
y = mx + c
So,
The given equation can be rewritten as:
y = 3x – 1
Now,
Compare the above equation with the standard form of the linear equation
y = mx + c
So,
By comparison,
We get
m = 3 and c = -1
Hence,
The representation of the given equation in the coordinate plane is:

Question 41.

Answer:

Question 42.
MATHEMATICAL CONNECTIONS
Graph the four equations in the same coordinate plane.
3y = -x – 3
2y – 14 = 4x
4x – 3 – y = 0
x – 12 = -3y
a. What enclosed shape do you think the lines form? Explain.
Answer:
The given equations are:
a. 3y = -x – 3
b. 2y – 14 = 4x
c. 4x – 3 – y = 0
d. x – 12 = -3y
Now,
Rewrite the given equations in the form of
y = mx + c
So,
a.
The given equation is:
3y = -x – 3
Divide by 3 into both sides
y = –\(\frac{x}{3}\) – (3 / 3)
y = –\(\frac{1}{3}\)x – 1

b.
The given equation is:
2y – 14 = 4x
Divide by 2 into both sides
y – (14 / 2) = 2x
y – 7 = 2x
y = 2x + 7

c.
The given equation is:
4x – 3 – y = 0
y = 4x – 3

d.
The given equation is:
x – 12 = -3y
Divide by -3 into both sides
y = –\(\frac{x}[3]\) + (12 / 3)
y = –\(\frac{1}{3}\)x + 4
Hence,
The representations of the given four equations in the coordinate plane is:

Hence,
By observing the graph,
We can conclude that the enclosed lines in the graph form a rectangle

b. Write a conjecture about the equations of parallel lines.
Answer:
We can determine from their equations whether two lines are parallel by comparing their slopes.
If the slopes are the same and the y-intercepts are different, the lines are parallel.
If the slopes are different, the lines are not parallel.
Unlike parallel lines, perpendicular lines do intersect.

Question 43.
MATHEMATICAL CONNECTIONS The graph shows the relationship between the width y and the length x of a rectangle in inches. The perimeter of a second rectangle is 10 inches less than the perimeter of the first rectangle.
a. Graph the relationship between the width and length of the second rectangle.

b. How does the graph in part (a) compare to the graph shown?
Answer:

Question 44.
MATHEMATICAL CONNECTIONS
The graph shows the relationship between the base length x and the side length (of the two equal sides) y of an isosceles triangle in meters. The perimeter of a second isosceles triangle is 8 meters more than the perimeter of the first triangle.

a. Graph the relationship between the base length and the side length of the second triangle.
The given graph is:

It is given that the graph shows the relationship between the base length x and the side length (of the two equal sides) y of an isosceles triangle in meters. The perimeter of a second isosceles triangle is 8 meters more than the perimeter of the first triangle.
Now,
From the graph,
The equation for the first Isosceles triangle is:
y = 6 – \(\frac{1}{2}\)x
Multiply by 2 into both sides
2y = 6 (2) – x
2y = 12 – x
2y + x = 12
The above equation represents the perimeter of the first Isosceles triangle
Where,
x is the base length
y is the side of the equal length like the third side
It is also given that the perimeter of a second isosceles triangle is 8 meters more than the perimeter of the first triangle.
So,
2y + x = 12 + 8
2y + x = 20
The above equation represents the perimeter of the second Isosceles triangle
Hence,
The representation of the perimeter of the second isosceles triangle in the coordinate plane is:

b. How does the graph in part (a) compare to the graph shown?
Answer:
The equation of the perimeter of the first isosceles triangle is:
y = 6 – \(\frac{1}{2}\)x
The equation of the perimeter of the second isosceles triangle is:
y = 10 – \(\frac{1}{2}\)x
Hence,
From comparing the above 2 equations with the standard form of the linear equation
y = mx + c
We can observe that the slopes are equal and only the y-intercepts are different
Hence, from the above,
We can conclude that the graphs of the perimeters of the first isosceles triangle and the second isosceles triangle are parallel

Question 45.
ANALYZING EQUATIONS
Determine which of the equations could be represented by each graph.

Answer:

Question 46.
MAKING AN ARGUMENT
Your friend says that you can write the equation of any line in slope-intercept form. Is your friend correct? Explain your reasoning.
Answer:
No, your friend is not correct

Explanation:
We can write the equation in slope-intercept form only when hat equation is in the linear form i.e., only x and y terms.
If there are exponential terms in an equation, then we can not write that equation in slope-intercept form
Hence, from the above,
We can conclude that your friend is not correct

Question 47.
WRITING
Write the definition of the slope of a line in two different ways.
Answer:

Question 48.
THOUGHT-PROVOKING
Your family goes on vacation to a beach 300 miles from your house. You reach your destination 6 hours after departing. Draw a graph that describes your trip. Explain what each part of your graph represents.
Answer:

Question 49.
ANALYZING A GRAPH
The graphs of the functions g(x) = 6x + a and h(x) = 2x + b, where a and b are constants, are shown. They intersect at the point (p, q).

a. Label the graphs of g and h.
b. What do a and b represent?
c. Starting at the point (p, q), trace the graph of g until you get to the point with the x-coordinate p + 2. Mark this point C. Do the same with the graph of h. Mark this point D. How much greater is the y-coordinate of point C than the y-coordinate of point D?
Answer:

Question 50.
HOW DO YOU SEE IT? Your commute to school by walking and by riding a bus. The graph represents your commute.

a. Describe your commute in words.
Answer:
The given graph is:

From the graph,
We can observe that your commute to school is different at different time intervals
Now,
From 0 to 10 seconds,
The distance you commuted gradually increases
From 10 to 14 seconds,
The distance you commuted is constant
From 14 to 18 seconds,
The distance you commuted abruptly increased

b. Calculate and interpret the slopes of the different parts of the graph.
Answer:
The given graph is:

From the graph,
We can observe that there are different commutes at different time intervals,
Now,
The x-axis represents the time
The y-axis represents the distance in miles
Now,
From 0 to 10 seconds,
The ordered pairs from the coordinate plane are:
(0, 0), (10, 0.5)
We know that,
The slope when 2 ordered pairs are given is represented as:
m = \(\frac{y2 – y1}{x2 – x1}\)
So,
From 0 to 10 seconds,
m = \(\frac{0.5 – 0}{10 – 0}\)
m = \(\frac{0.5}{10}\)
m = \(\frac{5}{100}\)
m = 0.05
From 10 to 14 seconds,
The ordered pairs from the coordinate plane are:
(10, 0.5), (14, 0.5)
We know that,
The slope when 2 ordered pairs are given is represented as:
m = \(\frac{y2 – y1}{x2 – x1}\)
So,
From 10 to 14 seconds,
m = \(\frac{0.5 – 0.5}{14 – 10}\)
m = \(\frac{0}{4}\)
m = 0
From 14 to 18 seconds,
The ordered pairs from the coordinate plane are:
(14, 0.5), (18, 2.5)
We know that,
The slope when 2 ordered pairs are given is represented as:
m = \(\frac{y2 – y1}{x2 – x1}\)
So,
From 14 to 18 seconds,
m = \(\frac{2.5 – 0.5}{18 – 14}\)
m = \(\frac{2}{4}\)
m = \(\frac{1}{2}\)
m = 0.5
Hence, fromthe above,
We can conclude that
The slope from 0 to 10 seconds is: 0.05
The slope from 10 to 14 seconds is: 0
The slope from 14 to 18 seconds is: 0.5

PROBLEM-SOLVING
In Exercises 51 and 52, find the value of k so that the graph of the equation has the given slope or y-intercept.

Question 51.
y = 4kx – 5; m = \(\frac{1}{2}\)
Answer:

Question 52.

Answer:
The given equation is:
y = –\(\frac{1}{3}\)x + \(\frac{5}{6}\)k
Compare the above equation with the standard representation of the linear equation.
We know that,
The standard representation of the linear equation is:
y = mx + b
On comparison,
We get,
m = –\(\frac{1}{3}\) and b = \(\frac{5}{6}\)k
It is given that
The value of b is: -10
So,
\(\frac{5}{6}\)k = -10
5k = -10 × 6
k = \(\frac{-10 × 6}{5}\)
k = -12
Hence, from the above,
We can conclude that the value of k is: -12

Question 53.
ABSTRACT REASONING
To show that the slope of a line is constant, let (x₁, y₁) and (x₂, y₂) be any two points on the line y = mx + b. Use the equation of the line to express y₁ in terms of x₁ and y₂ in terms of x₂. Then use the slope formula to show that the slope between the points is m.
Answer:

Maintaining Mathematical Proficiency

Find the coordinates of the figure after the transformation.

Question 54.
Translate the rectangle with 4 units left.

Answer:
The given graph is:

From the given graph,
The rectangle is covered up to 1 to 3 units at the x-axis and 0 to -4 units at the y-axis
So,
When we move the rectangle 4 units left, i.e., move the 1 and 3 units left for 4 units and move the 0 and -4 units left for 4 units
So,
The new rectangle in the graph is formed at:
At the x-axis:
1 – 4 = -3
3 – 4 = -1
At the y-axis:
0 – 4 = -4
-4 – 4 = -8
Hence,
The new rectangle is formed at (-1, -3) at the x-axis and at (-4, -8) at the y-axis
Hence,
The representation of the rectangle after the transformation in the coordinate plane is:

Question 55.
Dilate the triangle with respect to the origin using a scale factor of 2.

Answer:

Question 56.
Reflect the trapezoid in the y-axis.

Answer:
Determine whether the equation represents a linear or nonlinear function. Explain.

Question 57.
y – 9 = \(\frac{2}{x}\)
Answer:

Question 58.
x = 3 + 15y
Answer:
The given function is:
x = 3 + 15y
Rewrite the above given function in the form of
y = mx + c
So,
15y = 3 – x
y = \(\frac{3 – x}{15}\)
y = \(\frac{3}{15}\) – \(\frac{1}{15}\)x
Hence,
The above function is in the form of
y = mx + c
Hence, from the above,
We can conclude that the given function is linear

Question 59.
\(\frac{x}{4}\) + \(\frac{y}{12}\) = 1
Answer:

Question 60.
y = 3x⁴ – 6
Answer:
The given function is:
y = 3x⁴ – 6
Rewrite the above given function in the form of
y = mx + c
But,
The above function is not in the form of
y = mx + c
Hence, from the above,
We can conclude that the given function is non-linear

Lesson 3.6 Transformations of Graphs of Linear Functions

Essential Question
How does the graph of the linear function f(x) = x compare to the graphs of g(x) = f(x) + c and h(x) = f(cx)? Comparing Graphs of Functions
Answer:
The given linear function is:
f (x) = x
The graph corresponding to the above function will be the vertical line
Now,
The given functions are:
g (x) = f (x) + c and h (x) = f (cx)
Now,
The graph of g (x) will be the vertical line translated i.e., addition or subtraction by c units
The graph of h (x) will be the graph of the function f(x) stretched or compressed by 1/c units

EXPLORATION 1
Comparing Graphs of Functions
Work with a partner. The graph of f(x) = x is shown. Sketch the graph of each function, along with f, on the same set of coordinate axes. Use a graphing calculator to check your results. What can you conclude?
a. g(x) = x + 4
b. g(x) = x + 2
c. g(x) = x – 2
d. g(x) = x – 4


Answer:
The given functions are:
a. g(x) = x + 4
b. g(x) = x + 2
c. g(x) = x – 2
d. g(x) = x – 4
The other given function is:
f (x) = x
Now,
a.
The given function is:
g (x) = x + 4
Hence,
The representations of f(x) and g (x) in the same coordinate plane are:

Hence, from the above,
We can conclude that g (x) translated 4 units away from f(x) to the left side i.e., towards the positive y-axis

b.
The given function is:
g (x) = x + 2
Hence,
The representations of f(x) and g (x) in the same coordinate plane are:

Hence, from the above,
We can conclude that g (x) translated 2 units away from f(x) to the left side i.e., towards the positive y-axis

c.
The given function is:
g (x) = x – 2
Hence,
The representations of f(x) and g (x) in the same coordinate plane are:

Hence, from the above,
We can conclude that g (x) translated 2 units away from f(x) to the right side i.e., towards the negative y-axis

d,
The given function is:
g (x) = x – 4
Hence,
The representations of f(x) and g (x) in the same coordinate plane are:

Hence, from the above,
We can conclude that g (x) translated 4 units away from f(x) to the right side i.e., towards the negative y-axis

EXPLORATION 2
Comparing Graphs of Functions
Work with a partner. Sketch the graph of each function, along with f(x) = x, on the same set of coordinate axes. Use a graphing calculator to check your results. What can you conclude?
a. h(x) = \(\frac{1}{2}\)x
b. h(x) = 2x
c. h(x) = –\(\frac{1}{2}\)x
d. h(x) = -2x
Answer:
The given functions are:
a. h(x) = \(\frac{1}{2}\)x
b. h(x) = 2x
c. h(x) = –\(\frac{1}{2}\)x
d. h(x) = -2x
The other given function is:
f (x) = x
Now,
a.
The given function is:
h(x) = \(\frac{1}{2}\)x
Hence,
The representations of f (x) and h (x) in the same coordinate plane are:

Hence, from the above,
We can conclude that h (x) and f (x) passes through the origin and h (x) is steeper than f (x)

b.
The given function is:
h(x) = 2x
Hence,
The representation of f (x) and h(x) in the same coordinate plane is:

Hence, from the above,
We can conclude that h(x) and f(x) passes through the origin and f (x) is steeper than h (x)

c.
The given function is:
h(x) = –\(\frac{1}{2}\)x
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that h (x) and f (x) are on the opposite axes

d.
The given function is:
h(x) = -2x
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that f (x) and h (x) are in the opposite axes

EXPLORATION 3
Matching Functions with Their Graphs
Work with a partner. Match each function with its graph. Use a graphing calculator to check your results. Then use the results of Explorations 1 and 2 to compare the graph of k to the graph of f(x) = x.
a. k(x) = 2x – 4
b. k(x) = -2x + 2
c. k(x) = –\(\frac{1}{2}\)x + 4
d. k(x) = –\(\frac{1}{2}\)x — 2

Answer:
The given functions are:
a. k(x) = 2x – 4
b. k(x) = -2x + 2
c. k(x) = –\(\frac{1}{2}\)x + 4
d. k(x) = –\(\frac{1}{2}\)x — 2
The other given function is:
f (x) = x
Now,
a.
The given equation is:
k(x) = 2x – 4
Hence,
The representation of k (x) and f (x) in the coordinate plane is:

Hence, from the above,
We can conclude that graph C) matches k (x)
By comparing f (x) and k (x),
We can say that k (x) translates 2 units from f (x) towards the positive x-axis

b.
The given function is:
k(x) = -2x + 2
Hence,
The representation of k (x) and f (x) in the coordinate plane is:

Hence, from the above,
We can conclude that graph A) matches k (x)
By comparing f (x) and k (x),
We can say that k (x) and f (x) are on the opposite axes

c.
The given function is:
k(x) = –\(\frac{1}{2}\)x + 4
Hence,
The representation of k (x) and f (x) in the coordinate plane is:

Hence, from the above,
We can conclude that graph D) matches k (x)
By comparing f (x) and k (x),
We can say that k (x) and f (x) are on opposite axes and k (x) only translates through only the positive and negative x-axes

d.
The given function is:
k(x) = –\(\frac{1}{2}\)x — 2
Hence,
The representation of k (x) and f (x) in the coordinate plane is:

Hence, from the above,
We can conclude that graph B) matches k (x)
By comparing f (x) and k (x),
We can say that k (x) and f(x) are on the opposite axes and k (x) translates through only the negative x-axis and the negative y-axis

Communicate Your Answer

Question 4.
How does the graph of the linear function f(x) = x compare to the graphs of g(x) = f(x) + c and h(x) = f(cx)?
Answer:
The given linear function is:
f (x) = x
The graph corresponding to the above function will be the vertical line
Now,
The given functions are:
g (x) = f (x) + c and h (x) = f (cx)
Now,
The graph of g (x) will be the vertical line translated i.e., addition or subtraction by c units
The graph of h (x) will be the graph of the function f(x) stretched or compressed by 1/c units

3.6 Lesson

Monitoring Progress
Using f, graph (a) g and (b) h. Describe the transformations from the graph of f to the graphs of g and h.

Question 1.
f(x) = 3x + 1; g(x) = f(x) – 2; h(x) = f(x – 2)
Answer:
The given functions are:
f (x) = 3x + 1; g (x) = f (x) – 2 and h (x) = f (x – 2)
Now,
It is given that,
g (x) = f (x) – 2
So,
g (x) = 3x + 1 – 2
g (x) = 3x – 1
It is given that,
h (x) = f (x – 2)
h (x) = 3 (x – 2) + 1
h (x) = 3 (x) + 3 (2) + 1
h (x) = 3x + 6 + 1
h (x) = 3x + 7
Hence,
The representations of f (x), g (x), and h (x) in a coordinate plane is:

Hence, from the above,
We can conclude that
g (x) is translated 2 units away from f (x) toward the positive x-axis and h (x) is translated 5 units away from f (x) toward the positive y-axis

Question 2.
f(x) = -4x – 2; g(x) = -f(x); h(x) = f(-x)
Answer:
The given functions are:
f (x) = -4x – 2; g (x) = -f (x) and h (x) = f (-x)
Now,
It is given that,
g (x) = -f (x)
So,
g (x) = – (-4x – 2)
g (x) = 4x + 2
It is given that,
h (x) = f (-x)
h (x) = -4 (-x) – 2
h (x) = 4x – 2
Hence,
The representations of f (x), g (x), and h (x) in a coordinate plane is:

Hence, from the above,
We can conclude that
g (x) is translated 2.5 units away from f (x) toward the positive y-axis and h (x) is translated 1 unit away from f (x) toward the positive x-axis

Using f, graph (a) g and (b) h. Describe the transformations from the graph of f to the graphs of g and h.

Question 3.
f(x) = 4x – 2; g(x) = f (\(\frac{1}{2}\)x ); h(x) = 2f(x)
Answer:
The given functions are:
f (x) = 4x – 2; g (x) = f (\(\frac{1}{2}\)x ) and h (x) = 2f (x)
Now,
It is given that,
g (x) = f (\(\frac{1}{2}\)x )
So,
g (x) = 4 ( f (\(\frac{1}{2}\)x ) – 2
g (x) = 2x – 2
It is given that,
h (x) = 2f (x)
h (x) = 2 (4x – 2)
h (x) = 2 (4x) – 2(2)
h (x) = 8x – 4
Hence,
The representations of f (x), g (x), and h (x) in a coordinate plane is:

Hence, from the above,
We can conclude that
g (x) is translated 0.5 units away from f (x) toward the positive x-axis and h (x) is translated 0 units away from f (x) toward the origin

Question 4.
f(x) = -3x + 4; g(x) = f(2x); h(x) = \(\frac{1}{2}\)f(x)
Answer:
The given functions are:
f (x) = -3x + 4; g (x) = f (2x) and h (x) = \(\frac{1}{2}\)f(x)
Now,
It is given that,
g (x) = f (2x)
So,
g (x) = -3 (2x) + 4
g (x) = -6x + 4
It is given that,
h (x) = \(\frac{1}{2}\)f(x)
h (x) = \(\frac{1}{2}\) (3x + 4)
h (x) = \(\frac{3}{2}\)x + \(\frac{4}{2}\)
h (x) = \(\frac{3}{2}\)x + 2
Hence,
The representations of f (x), g (x), and h (x) in a coordinate plane is:

Hence, from the above,
We can conclude that
g (x) and f (x) are on the same axis and h (x) and f (x) are on the opposite axis

Question 5.
Graph f(x) = x and h(x) = \(\frac{1}{2}\)x – 2. Describe the transformations from the graph of f to the graph of h.
Answer:
The given functions are:
f (x) = x and h (x) = \(\frac{1}{2}\)x – 2
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that h (x) translated 4 units away from f (x) towards the positive x-axis

Transformations of Graphs of Linear Functions 3.6 Exercises

Vocabulary and Core Concept Check

Question 1.
WRITING
Describe the relationship between f(x) = x and all other nonconstant linear functions.
Answer:

Question 2.
VOCABULARY
Name four types of transformations. Give an example of each and describe how it affects the graph of a function.
Answer:
The four types of transformations that affect the graph of a function are:
a. Translation
Ex:
f (x) = x and g (x) = 2 f(x)
b. Rotation
Ex:
f (x) = 3x + 2 and g (x) = 3x – 2
c. Reflection
Ex:
f (x) =x and g (x) =-f (x)
d. Dilation
Ex:
f (x) = 3x + 6 and g (x) = f (2x)

Question 3.
WRITING
How does the value of a in the equation y = f(ax) affect the graph of y = f(x)? How does the value of a in the equation y = af(x) affect the graph of y = f(x)?

Answer:

Question 4.
REASONING
The functions f and g are linear functions. The graph of g is a vertical shrink of the graph of f. What can you say about the x-intercepts of the graphs of f and g? Is this always true? Explain.
Answer:
It is given that the functions f and g are linear functions
We know that,
f can be written as f (x)
g can be written as g (x)
It is also given that the graph of g is a vertical shrink of the graph of f.
So,
Since the graph of g shrinks, then the x-intercept of g will also shrink if we observe the functions of f and g
Hence, from the above,
We can conclude that the x-intercept of g will shrink and this is always true

In Exercises 5–10, use the graphs of f and g to describe the transformation from the graph of f to the graph of g.

Question 5.

Answer:

Question 6.

Answer:
The given graph is:

From the above graph,
The given functions are:
f (x) = x – 3
g (x) = f (x + 4)
So,
g (x) = (x + 4) – 3
g (x) = x + 1
Hence,
When we observe f (x) and g (x), we can say that
g (x) shrinks by 2 units of f (x)

Question 7.
f(x) = \(\frac{1}{3}\)x + 3; g(x) = f(x) = -3
Answer:

Question 8.
f(x) = -3x + 4; g(x) = f(x) + 1
Answer:
The given functions are:
f (x) = -3x + 4
g (x) = f (x) + 1
So,
g (x) = -3x + 4 + 1
g (x) = -3x + 5
Hence,
The representation of f and g in the same coordinate plane is:

Hence, from the above,
We can conclude that g (x) translates 1 unit away from f (x)

Question 9.
f(x) = -x – 2; g(x) = f(x + 5)
Answer:

Question 10.
f(x) = \(\frac{1}{2}\)x – 5; g(x) = f(x – 3)
Answer:
The given functions are:
f (x) = \(\frac{1}{2}\)x – 5
g (x) = f (x – 3)
So,
g (x) = \(\frac{1}{2}\) ( x – 3 ) – 5
Hence,
The representation of f (x) and g (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that g (x) is dilated from f (x)

Question 11.
MODELING WITH MATHEMATICS
You and a friend start biking from the same location. Your distance d (in miles) after t minutes is given by the function d(t) = \(\frac{1}{5}\)t. Your friend starts biking 5 minutes after you. Your friend's distance f is given by the function f(t) = d(t – 5). Describe the transformation from the graph of d to the graph of f.

Answer:

Question 12.
MODELING WITH MATHEMATICS
The total cost C (in dollars) to cater an event with p people is given by the function C(p) = 18p + 50. The set-up fee increases by $25. The new total cost T is given by the function T(p) = C(p) + 25. Describe the transformation from the graph of C to the graph of T.

Answer:
It is given that the total cost C (in dollars) to cater an event with p people is given by the function
C(p) = 18p + 50
It is also given that the set-up fee increases by $25.
So,
The new total cost T is given by the function
T(p) = C(p) + 25
So,
T (p) = 18p + 50 + 25
T (p) = 18p + 75
Hence,
The representation of T (p) and C (p) in the same coordinate plane is:

Hence, from the above,
We can conclude that T (p) translated 25 units away from C (p)

In Exercises 13–16, use the graphs of f and h to describe the transformation from the graph of f to the graph of h.

Question 13.

Answer:

Question 14.

Answer:
The given graph is:

From the above graph,
The given functions are:
f (x) = -3x + 1
h (x) = f (-x)
So,
h (x) = -3 (-x) + 1
h (x) = 3x + 1
Hence,
The representations of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that f (x) and h (x) are perpendicular lines

Question 15.
f(x) = -5 – x; h(x) = f(-x)
Answer:

Question 16.
f(x) = \(\frac{1}{4}\)x – 2; h(x) = -f(x)
Answer:
The given functions are:
f (x) = \(\frac{1}{4}\)x – 2
h (x) = – f(x)
So,
h (x) = – (\(\frac{1}{4}\)x – 2)
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that h (x) is a reflection of f (x) on the y-axis

In Exercises 17–22, use the graphs of f and r to describe the transformation from the graph of f to the graph of r.

Question 17.

Answer:

Question 18.

Answer:
The given graph is:

From the above graph,
The given functions are:
f (x) = -x
r (x) = f (\(\frac{1}{4}\) x)
So,
r (x) = –\(\frac{1}{4}\)x
Hence,
The representation of f (x) and r (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that r (x) shrinks by \(\frac{1}{4}\) of f (x)

Question 19.
f(x) = -2x – 4; r(x) = f(\(\frac{1}{2}\)x )
Answer:

Question 20.
f(x) = 3x + 5; r(x) = f (\(\frac{1}{3}\)x)
Answer:
The given functions are:
f (x) = 3x + 5
r (x) = f (\(\frac{1}{3}\)x)
So,
r (x) = 3 ( \(\frac{1}{3}\) x ) + 5
r (x) = x + 5
Hence,
The representation of f (x) and r (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that r (x) shrinks by 1 unit of f (x)

Question 21.
f(x) = \(\frac{2}{3}\)x + 1; r(x) = 3f(x)
Answer:

Question 22.
f(x) = –\(\frac{1}{4}\)x – 2; r(x) = 4f(x)
Answer:
The given functions are:
f (x) = –\(\frac{1}{4}\)x – 2
r (x) = 4 f (x)
So,
r (x) = 4 [-\(\frac{1}{4}\)x – 2]
r (x) = -x – 8
Hence,
The representation of f (x) and r (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that r (x) translates 4 units away from  f (x)

In Exercises 23–28, use the graphs of f and h to describe the transformation from the graph of f to the graph of h.

Question 23.

Answer:

Question 24.

Answer:
The given graph is:

From the above grah,
The given functions are:
f (x) = -2x – 6
h (x) = \(\frac{1}{3}\) f(x)
So,
h (x) = –\(\frac{1}{3}\) (2x + 6)
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that h (x) stretches by \(\frac{1}{3}\) of f (x)

Question 25.
f(x) = 3x – 12; h(x) = \(\frac{1}{6}\)f(x)
Answer:

Question 26.
f(x) = -x + 1; h(x) = f(2x)
Answer:
The given functions are:
f (x) = -x + 1
h (x) = f (2x)
So,
h (x) = -(2x) + 1
h (x) = -2x + 1
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that h (x) stretches by 1 unit from f (x)

Question 27.
f(x) = -2x – 2; h(x) = f(5x)
Answer:

Question 28.
f(x) = 4x + 8; h(x) = \(\frac{3}{4}\)f(x)
Answer:
The given functions are:
f (x) = 4x + 8
h (x) = \(\frac{3}{4}\) f (x)
So,
h (x) = \(\frac{3}{4}\) (4x + 8)
h (x) = \(\frac{3}{4}\) (4x) + \(\frac{3}{4}\) (8)
h (x) = 3x + 6
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that h (x) shrinks by 2 units from f (x)

In Exercises 29–34, use the graphs of f and g to describe the transformation from the graph of f to the graph of g.

Question 29.
f(x) = x – 2; g(x) = \(\frac{1}{4}\)f(x)
Answer:

Question 30.
f(x) = -4x + 8; g(x) = -f(x)
Answer:
The given functions are:
f (x) = -4x + 8
g (x) = -f (x)
So,
g (x) = -4 (-x) + 8
g (x) = 4x + 8
Hence,
The representation of f (x) and g (x) in the same coordinate plane is:

Hence, from the above graph,
We can conclude that g (x) is a reflection of f (x)

Question 31.
f(x) = -2x – 7; g(x) = f(x – 2)
Answer:

Question 32.
f(x) = 3x + 8; g(x) = f(\(\frac{2}{3}\)x)
Answer:
The given functions are:
f (x) = 3x + 8
g (x) = f (\(\frac{2}{3}\)x)
So,
g (x) = 3 ( \(\frac{2}{3}\) ) x + 8
g (x) = 2x + 8
Hence,
The representation of f (x) and g (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that g (x) shrink by 1 unit of the graph of f (x)

Question 33.
f(x) = x – 6; g(x) = 6f(x)
Answer:

Question 34.
f(x) = -x; g(x) = f(x) -3
Answer:
The given functions are:
f (x) = -x
g (x) = f (x) – 3
So,
g (x) = -x – 3
Hence,
The representation of f (x) and g (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that g (x) is translated by 3 units away from f (x)

In Exercises 35–38, write a function g in terms of f so that the statement is true.

Question 35.
The graph of g is a horizontal translation 2 units right of the graph of f.
Answer:

Question 36.
The graph of g is a reflection in the y-axis of the graph of f.
Answer:
We know that,
f can be writen as f (x)
g can be written as g (x)
Now,
The given statement is:
The graph of g is a reflection of f(x) in the y-axis of the graph of f (x)
So,
Reflection in the y-axis means if f (x) is +ve, then the reflection of f (x) will be -ve and vice-versa
Hence,
The representation of the given statement in terms of f(x) is:
g (x) = f (-x)

Question 37.
The graph of g is a vertical stretch by a factor of 4 of the graph of f.
Answer:

Question 38.
The graph of g is a horizontal shrink by a factor of \(\frac{1}{5}\) of the graph of f.
Answer:
We know that,
f can be writen as f (x)
g can be written as g (x)
Now,
The given statement is:
The graph of g is a horizontal shrink by a factor of \(\frac{1}{5}\) of the graph
We know that,
"Shrink" is represented by '-'
"Stretch" is represented by '+'
Hence,
The representation of the given statement in terms of f (x) is:
g (x) = – \(\frac{1}{5}\) f (x)

ERROR ANALYSIS In Exercises 39 and 40, describe and correct the error in graphing g.

Question 39.

Answer:

Question 40.

Answer:
The given functions are:
f (x) = -x + 3 and g (x) = f (-x)
So,
g (x) = -[-x] + 3
g (x) = x + 3
Hence,
The representations of f (x) and g (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that f (x) and g (x) are perpendicular lines from the graph

In Exercises 41–46, graph f and h. Describe the transformations from the graph of f to the graph of h.

Question 41.
f(x) = x; h(x) = \(\frac{1}{3}\)x + 1
Answer:

Question 42.
f(x) = x; h(x) = 4x – 2
Answer:
The given functions are:
f (x) = x and h (x) = 4x – 2
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that the transformations are a vertical stretch by a factor of 4 followed by a vertical translation of 2 units down

Question 43.
f(x) = x; h(x) = -3x – 4
Answer:

Question 44.
f(x) = x; h(x) = –\(\frac{1}{2}\)x + 3
Answer:
The given functions are:
f (x) = x and h (x) = –\(\frac{1}{2}\)x + 3
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that the transformations are a vertical shrink by a factor of \(\frac{1}{2}\) followed by a vertical translation of 3 units up

Question 45.
f(x) = 2x; h(x) = 6x – 5
Answer:

Question 46.
f(x) = 3x; h(x) = -3x – 7
Answer:
The given functions are:
f (x) = 3x and h (x) = -3x – 7
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that the transformations are a vertical shrink by a factor of 3 followed by a vertical translation of 7 units down

Question 47.
MODELING WITH MATHEMATICS
The function t(x) = -4x +72 represents the temperature from 5 P.M. to 11 P.M., where x is the number of hours after 5 P.M. The function d(x) = 4x + 72 represents the temperature from 10 A.M. to 4 P.M., where x is the number of hours after 10 A.M. Describe the transformation from the graph of t to the graph of d.

Answer:

Question 48.
MODELING WITH MATHEMATICS
A school sells T-shirts to promote school spirit. The school's profit is given by the function P(x) = 8x – 150, where x is the number of T-shirts sold. During the playoffs, the school increases the price of the T-shirts. The school's profit during the play-offs is given by the function Q(x) = 16x – 200, where x is the number of T-shirts sold. Describe the transformations from the graph of P to the graph of Q.

Answer:
It is given that a school sells T-shirts to promote school spirit. The school's profit is given by the function
P(x) =8x- 150
where,
x is the number of T-shirts sold
It is also given that during the playoffs, the school increases the price of the T-shirts. The school's profit during the play-offs is given by the function
Q(x) = 16x – 200
where,
x is the number of T-shirts sold.
Hence,
The representation of P (x) and Q(x) in the same coordinate plane is:

Hence, from the above graph,
We can conclude that
The x-axis shrink by a factor of 2
The y-axis shrink by 50

Question 49.
USING STRUCTURE
The graph of g(x) = a • f(x – b) + c is a transformation of the graph of the linear function f. Select the word or value that makes each statement true.

a. The graph of g is a vertical ______ of the graph of f when a = 4, b = 0, and c = 0.
b. The graph of g is a horizontal translation ______ of the graph of f when a = 1, b = 2, and c = 0.
c. The graph of g is a vertical translation 1 unit up of the graph of f when a = 1, b = 0, and c = ____.
Answer:

Question 50.
USING STRUCTURE
The graph of h(x) = a • f(bx – c) + d is a transformation of the graph of the linear function f. Select the word or value that makes each statement true.

a. The graph of h is a ______ shrink of the graph of f when a = \(\frac{1}{3}\), b = 1, c = 0, and d = 0.
b. The graph of h is a reflection in the ______ of the graph of f when a = 1, b = -1, c = 0, and d = 0.
c. The graph of h is a horizontal stretch of the graph of f by a factor of 5 when a = 1, b = _____, c = 0, and d = 0.
Answer:
The given original function is:
f (x) = x
The given choice of words to make the given statements true is:

The transformation of the function of f (x) is given as:
h (x) = a f (bx – c) + d
Where,
a, b, c, and d are constants
Now,
a.
The given values of constants are:
a = \(\frac{1}{3}\),
b = 1
c = 0
d = 0
So,
h (x) = \(\frac{1}{3}\)f (1 (x) – 0) + 0
h (x) = \(\frac{1}{3}\) f (x)
So,
h (x) = \(\frac{1}{3}\)x
Hence,
The representations of f (x) and h (x) in the coordinate plane is:

Hence, from the above,
We can conclude that the graph of h is a vertical shrink of the graph of f when a = \(\frac{1}{3}\), b = 1, c = 0, and d = 0.

b.
The given values of constants are:
a = 1
b = -1
c = 0
d = 0
So,
h (x) = 1f (-1 (x) – 0) + 0
h (x) =  f (-x)
So,
h (x) = -x
Hence,
The representations of f (x) and h (x) in the coordinate plane is:

Hence, from the above,
We can conclude that the graph of h is a reflection in the y-axis of the graph of f when a = 1, b = -1, c = 0, and d = 0

c.
The given values of constants are:
a = 1
b = p
c = 0
d = 0
So,
h (x) = 1f (p (x) – 0) + 0
h (x) =  f (px)
So,
h (x) = px
Hence,
The representations of f (x) and h (x) in the coordinate plane is:

Hence, from the above,
We can conclude that the graph of h is a horizontal stretch of the graph of f by a factor of 5 when a = 1, b = \(\frac{1}{5}\), c = 0, and d = 0

Question 51.
ANALYZING GRAPHS
Which of the graphs are related by only a translation? Explain.

Answer:

Question 52.
ANALYZING RELATIONSHIPS
A swimming pool is filled with water by a hose at a rate of 1020 gallons per hour. The amount v (in gallons) of water in the pool after t hours is given by the function v(t) = 1020t. How does the graph of v change in each situation?
a. A larger hose is found. Then the pool is filled at a rate of 1360 gallons per hour.
Answer:
It is given that a swimming pool is filled with water by a hose at a rate of 1020 gallons per hour. The amount v (in gallons) of water in the pool after t hours is given by the function
v(t) = 1020t
Now,
It is also given that a pool is filled at a rate of 1360 gallons per hour
So,
The overall rate change = 1340 / 1020
= 440 / 330
= 4 / 3
So,
The new rate of change is representad by the function
f (t) = (4 / 3) v (t)
Hence,
The representation of the graph of v using the functions v (t) = 1020t and v (t) = 1360t is:

b. Before filling up the pool with a hose, a water truck adds 2000 gallons of water to the pool.
Answer:
We know that,
From part (a),
The function for filling up the pool with a hose is given as:
v (t) = 1020t
It is given that before filling up the pool with a hose, a water truck adds 2000 gallons of water to the pool
So,
v (t) = 1020t + 2000
Hence,
The representations of the functions before filling the hose and  a water truck adds 2000 gallons of water are:

Question 53.
ANALYZING RELATIONSHIPS
You have $50 to spend on fabric for a blanket. The amount m (in dollars) of money you have after buying y yards of fabric is given by the function m(y) = -9.98y + 50. How does the graph of m change in each situation?

a. You receive an additional $10 to spend on the fabric.
b. The fabric goes on sale, and each yard now costs $4.99.
Answer:

Question 54.
THOUGHT-PROVOKING
Write a function g whose graph passes through the point (4, 2) and is a transformation of the graph of f(x) = x.
Answer:
It is given that a function g whose graph passes through the point (4, 2) and is a transformation of the graph of f(x) = x
We know that,
The equation that passes through (h, k) and it is a transformation of another function is:
y = fx – h) + k
So,
From (4, 2)
As the translation between the values of x and y in the given point is 2,
We can make the value of 'h' as -2
We can make the value of k as -4
Hence,
The equation that passes through (4, 2) is:
y = (x + 2 ) – 4
y = x – 2
Hence, from the above,
We can conclude that the function that passes through (4, 2) is:
g (x) = x – 2

In Exercises 55–60, graph f and g. Write g in terms of f. Describe the transformation from the graph of f to the graph of g.

Question 55.
f(x) = 2x – 5; g(x) = 2x – 8
Answer:

Question 56.
f(x) = 4x + 1; g(x) = -4x – 1
Answer:
The given equations are:
f (x) = 4x + 1
g (x) = -4x – 1
So,
g (x) = – (4x + 1)
g (x) = -f (x)
Hence,
The representations of f (x) and g (x) in the coordinate plane is:

Hence, from the above,
We can conclude that f (x) and g (x) are on the opposite quadrants

Question 57.
f(x) = 3x + 9; g(x) = 3x + 15
Answer:

Question 58.
f(x) = -x – 4 ; g(x) = x – 4
Answer:
The given equations are:
f (x) = -x – 4
g (x) = x – 4
Hence,
The representations of f (x) and g (x) in the coordinate plane are:

Hence, from the above,
We can conclude that f (x) and g (x) are perpendicular lines since only the slopes vary and the y-intercepts are the same

Question 59.
f(x) = x + 2; g(x) = \(\frac{2}{3}\)x + 2
Answer:

Question 60.
f(x) = x – 1; g(x) = 3x – 3
Answer:
The given equations are:
f (x) = x – 1
g (x) = 3x – 3
So,
g (x) = 3 (x – 1)
g (x) = 3 f (x)
Hence,
The representations of f (x) and g (x) in the coordinate plane are:

Hence, from the above,
We can conclude that g (x) translates 3 units away from f (x)

Question 61.
REASONING
The graph of f(x) = x + 5 is a vertical translation 5 units up of the graph of f(x) = x. How can you obtain the graph of f(x) = x + 5 from the graph of f(x) = x using a horizontal translation?
Answer:

Question 62.
HOW DO YOU SEE IT? Match each function with its graph. Explain your reasoning.

a. a(x) = f(-x)
b. g(x) = f(x) – 4
c. h(x) = f(x) + 2
d. k(x) = f(3x)
Answer:

REASONING
In Exercises 63–66, find the value of r.

Question 63.

Answer:

Question 64.

Answer:
The given equations are:
f (x) = -3x + 5
g (x) = f (rx)
So,
g (x) = -3 (rx) + 5
Where,
r is the transformational factor
From the graph,
The x-intercept of g(x) is: 5
Now,
To find he x-intercept, put y = 0
So,
0 = -3 (rx) + 5
3 (rx) = 5
3r (5) = 5
3r = 5 / 5
3r = 1
r = \(\frac{1}{3}\)
Hence, from the above,
WE can conclude that the value of r is: \(\frac{1}{3}\)

Question 65.

Answer:

Question 66.

Answer:
The given equations are:
f (x) = \(\frac{1}{2}\)x + 8
g (x) = f (x) + r
So,
g (x) = \(\frac{1}{2}\)x + 8 + r
From the graph,
We can observe that f (x) and g (x) are both the parallel lines
So,
The slopes of f (x) and g (x) are the same and only the y-intercepts differ in f (x) and g (x)
From the graph,
We can observe that the y-intercept of g (x) is: 0
So,
g (x) = \(\frac{1}{2}\)x + 8 + 0
Hence, from the above,
We can conclude that the value of r is: 0

Question 67.
CRITICAL THINKING
When is the graph of y = f(x) + w the same as the graph of y = f(x + w) for linear functions? Explain your reasoning.
Answer:

Maintaining Mathematical Proficiency

Solve the formula for the indicated variable.(Section 1.5)

Question 68.
Solve for h.

Answer:
The given formula from the given figure is:
V = πr²h
Divide by πr² into both sides
So,
\(\frac{V}{πr²}\) = h
Hence, from the above,
We can conclude that the formula for 'h' is:
h = \(\frac{V}{πr²}\)

Question 69.
Solve for w.

Answer:

Solve the inequality. Graph the solution, if possible. (Section 2.6)

Question 70.
| x – 3 | ≤ 14
Answer:
The given absolute value inequality is:
| x + 3 | ≤ 14
We know that,
| x | = x for x > 0
| x | = -x for x < 0
So,
x + 3 ≤ 14 and x + 3 ≥ -14
x ≤ 14 – 3 and x ≥ -14 – 3
x ≤ 11 and x ≥ -17
Hence, from the above,
We can conclude that the solutions to the given absolute value inequality are:
x ≤11 and x ≥ -17
The representation of the solutions of the given inequality in the graph is:

Question 71.
| 2x + 4 | > 16
Answer:

Question 72.
5 | x + 7 | < 25
Answer:
The given absolute value inequality is:
5 | x + 7 | < 25
So,
| x + 7 | < 25 / 5
| x + 7 | < 5
We know that,
| x | = x for x > 0
| x | = -x for x < 0
So,
x + 7 < 5 and x + 7 > -5
x < 5 – 7 and x > -5 – 7
x < -2 and x > -12
Hence, from the above,
We can conclude that the solutions of the given absolute value inequality are:
x < -2 and x > -12
The representations of the solutions of the given absolute value inequality in the graph is:

Question 73.
-2 | x + 1 | ≥ 18
Answer:

Lesson 3.7 Graphing Absolute Value Functions

Essential Question
How do the values of a, h, and k affect the graph of the absolute value function g(x) = a | x – h | + k? The parent absolute value function is f(x) = | x | Parent absolute value function
The graph of f is V-shaped.
Answer:
The given absolute value function is:
g (x) = a | x – h | + k
The parent function for the given absolute value function is:
f (x) = | x |
It is also given that the graph of f is v-shaped
Now,
The given absolute value function has the constants a, x, h, and k
Now,
"x – h" represents the translation of the x-axis where h is the translation value
Ex:
x – h =x – 3
Where,
x is the original function and h is the translation factor of x
Now,
"k" represents the y-intercept of the given absolute value function
The value of k affects the graph of the given absolute value function to translate on the y-axis up and down or shrink and stretch

EXPLORATION 1
Identifying Graphs of Absolute Value Functions
Work with a partner. Match each absolute value function with its graph. Then use a graphing calculator to verify your answers.
a. g(x) = | x – 2 |
b. g(x) = | x – 2 | + 2
c. g(x) = | x + 2 | + 2
d. g(x) = – | x + 2 | + 2
e. g(x) = 2 | x – 2 |
f. g(x) = – | x + 2 | + 2

Answer:
The given absolute value functions are:
a. g(x) = | x – 2 |
b. g(x) = | x – 2 | + 2
c. g(x) = | x + 2 | + 2
d. g(x) = – | x + 2 | + 2
e. g(x) = 2 | x – 2 |
f. g(x) = – | x + 2 | + 2
We know that,
| x | =x for x > 0
| x | = -x for x < 0
Now,
a.
The given absolute value function is:
g(x) = | x – 2 |
So,
g (x) = x – 2 or g (x) = – (x – 2)
g (x) = x – 2 or g (x) = 2 – x
Hence,
The representation of the given absolute value function in the coordinate plane is:

Hence, fro the above,
We can conclude that graph F) matches the given absolute value function
b.
The given absolute value function is:
g(x) = | x – 2 | + 2
So,
g (x) = x – 2 + 2 or g (x) = – (x – 2) + 2
g (x) = x or g (x) = 2 – x + 2
g (x) = x or g (x) = 4 – x
Hence,
The representation of the given absolute value function in the coordinate plane is:

Hence, from the above,
We can conclude that graph C) matches the given absolute value function
c.
The given absolute value function is:
g(x) = | x + 2 | + 2
So,
g (x) = x + 2 + 2 or g (x) = – (x + 2 ) + 2
g (x) = x + 4 or  g (x) = -x – 2 + 2
g (x) = x + 4 or g (x) = -x
Hence,
The representation of the given absolute value function in the coordinate plane is:

Hence, from the above,
We can conclude that graph C) matches the given absolute value function
d.
The given absolute value function is:
g(x) = – | x + 2 | + 2
So,
g (x) = – (x + 2 )  + 2 or g (x) = – [- (x + 2 ) ] + 2
g (x) = -x – 2 + 2 or g (x) = x + 2 + 2
g (x) = x or g (x) =x + 4
Hence,
The representation of the given absolute value function in the coordinate plane is:

Hence, from the above,
We can conclude that graph B) matches the given absolute value function
e.
The given absolute value function is:
g(x) = 2 | x – 2 |
So,
g (x) = 2 (x – 2) or g (x) = 2 ( – (x – 2) )
g (x) = 2x – 4 or g (x) = 2 (-x + 2 )
g (x)= 2x – 4 or g (x) = -2x + 4
Hence,
The representation of the given absolue value functions in the coordinate plane is:

Hence, from the above,
We can conclude that the graph A) matches the given absolute value function
f.
The given absolute value function is:
g(x) = – | x + 2 | + 2
So,
g (x) = – (x + 2) + 2 or g (x) = – [- (x – 2} ] + 7
g (x) = x + 4 or g (x) = 2 – x + 7
g (x) = x + 4 or g (x) = -x + 9
Hence,
The representation of the given absolute value function in the coordinate plane is:

Hence, from the above,
We can conclude that graph E) matches the given absolute value function

Communicate Your Answer

Question 2.
How do the values of a, h, and k affect the graph of the absolute value function g(x) = a | x – h | + k?

Answer:
The given absolute value function is:
g (x) = a | x – h | + k
From the above absolute value function,
We can say that
a, h and k are constants
Now,
"a" represents the constant to multiply g (x)
"h" represents the translation factor on the x-axis
"k" represents the y-intercept of the given function
The value of "h" and "k" affect the graph in such a way that the graph can move anywhere in the graph

Question 3.
Write the equation of the absolute value function whose graph is shown. Use a graphing calculator to verify your equation.

Answer:
The given graphing calculator is:

From the above graph,
We can observe that the plot is in downwqrd direction i.e., the absolute value fuction is negative and the plot passes through 2 in the x-axis
Hence,
The absolute value function representing the given graph is:
f (x) = -| x + 1 | + 2

3.7 Lesson

Monitoring Progress

Graph the function. Compare the graph to the graph of f(x) = | x |. Describe the domain and range.

Question 1.
h(x) = | x | – 1
Answer:
The given functions are:
f (x) = | x |
h (x) = | x | – 1
Hence,
The representation of f (x) and h (x) in  the same coordinate plane is:

Hence, from the above graph,
h (x) is translated vertically away from f (x) on the y-axis by 1unit
Now,
The domain of the given functions are: -10 ≤ x ≤ 10
The range of the given functions are: -1 ≤ y ≤ 10

Question 2.
n(x) = | x + 4 |
Answer:
The given functions are:
f (x) = | x |
h (x) = | x + 4 |
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above graph,
h (x) is translated from f (x) 4 units away on the x-axis
Now,
The domain of f (x) is: -10 ≤ x ≤ 10
The domain of h (x) is: -10 ≤ x ≤6
The range of f (x) is: 0 ≤ y ≤ 10
The range of h (x) is: 0 ≤y ≤ 6

Graph the function. Compare the graph to the graph of f(x) = | x |. Describe the domain and range.

Question 3.
t(x) = -3| x |
Answer:
The given functions are:
f (x) = | x |
t (x) = -3 | x |
Hence,
The representation of f (x) and t (x) in the same coordinate plane is:

Hence, from the above graph,
t (x) is shrunk by 3units and rotated by 90° on the y-axis
Now,
The domain of f (x) is: -10 ≤ x ≤ 10
The domain of t (x) is: -3 ≤x ≤ 3
The range of f (x) is: 0 ≤ y ≤ 10
The range of t (x) is: -10 ≤ y ≤ 0

Question 4.
v(x) = \(\frac{1}{4}\)| x |
Answer:
The given functions are:
f (x) = | x |
v (x) = \(\frac{1}{4}\)| | x |
Hence,
The representation of f (x) and v (x) in the same coordinate plane is:

Hence, from the above graph,
t (x) translates \(\frac{1}{4}\)| units of f (x) on the y-axis
Now,
The domain of the given functions are: -10 ≤ x ≤ 10
The range of f (x) is: 0 ≤ y ≤ 10
The range of t (x) is: 0 ≤ y ≤ 2.5

Question 5.
Graph f(x) = | x – 1 | and g(x) = | \(\frac{1}{2}\)x – 1 |.
Compare the graph of g to the graph of f.
Answer:
The given functions are:
f (x) = | x – 1 |
g (x) = | \(\frac{1}{2}\)x – 1 |
Hence,
The representation of f (x) and g (x) in the same coordinate plane is:

Hence, from the above graph,
g (x) translates 1 unit away from f (x) on the x-axis
Now,
The domain of f (x) is: -8 ≤ x ≤  10
The domain of g (x) is: -10 ≤  x ≤  10
The range of f (x) is: 0 ≤  y ≤ 10
The range of g (x) is: 0 ≤  y ≤  6

Question 6.
Graph f(x) = | x + 2 | + 2 and g(x) = | -4x + 2 | + 2.
Compare the graph of g to the graph of f.
Answer:
The given functions are:
f (x) = | x + 2 | + 2
g (x) = | -4x + 2 | + 2
Hence,
The representation of f (x) and g(x) in the same coordinate plane is:

Hence, from the above graph,
g (x) translates \(\frac{3}{2}\) units away from f (x) on the x-axis
Now,
The domain of f (x) is: -10 ≤ x ≤ 6
The domain of g (x) is:-1.5 ≤ x ≤ 2.5
The range of f (x) is: 0 ≤ y ≤ 10
The range of g (x) is: 2 ≤y ≤ 10

Question 7.
Let g(x) = | –\(\frac{1}{2}\)x + 2 | + 1.
(a) Describe the transformations from the graph of f(x) = | x | to the graph of g.
(b) Graph g.
Answer:
a.
The given functions are:
f (x) = | x |
g (x) = | –\(\frac{1}{2}\)x + 2 | + 1
Hence,
The representation of f (x) and g (x) in the same coordinate plane is:

Hence, from the above graph,
We can observe that
g (x) translates 4 units away from f (x) on the positive x-axis
g (x) translates 2 units away from f (x) on the negative x-axis

b.
The function of g (x) is:
| –\(\frac{1}{2}\)x + 2 | + 1
We know that,
g can be written as g (x)
Hence,
The representation of g (x) in the coordinate plane is:

Graphing Absolute Value Functions 3.7 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
The point (1, -4) is the _______ of the graph of f(x) = -3 | x – 1 | – 4.
Answer:

Question 2.
USING STRUCTURE
How do you know whether the graph of f(x) = a | x – h | + k is a vertical stretch or a vertical shrink of the graph of f(x) = | x | ?
Answer:
The given function is:
f (x) = a | x – h | + k
Where,
a is the vertical stretch or vertical shrink
h is the horizontal translation
k is the vertical shift
Now,
'a' will be the vertical stretch if the value of a is an integer
'a' will be the vertical shrink if the value of 'a' is a fraction
Hence, from the above,
We can conclude that the given function is a vertical stretch

Question 3.
WRITING
Describe three different types of transformations of the graph of an absolute value function.
Answer:

Question 4.
REASONING
The graph of which function has the same y-intercept as the graph of f(x) = | x – 2 | + 5? Explain.

Answer:
The given absolute value functions are:
a. f (x) = | x – 2 | + 5
b. g (x) = | 3x – 2 | + 5
c. h (x) = 3 | x – 2 | + 5
We know that,
| x | = x for x > 0
| x | = -x for x < 0
Now,
a.
The given absolute value function is:
f (x) = | x – 2 | + 5
So,
f (x) = x – 2 + 5 or f (x) = -(x – 2) + 5
f (x) = x + 3 or f (x) = -x + 2 + 5
f (x) = x + 3 or f (x) = -x + 7

b.
The given absolute value function is:
g (x) = | 3x – 2 | + 5
So,
g (x) = 3x – 2 + 5 or g (x) = – (3x – 2 ) + 5
g (x) = 3x + 3 or g (x) = -3x + 2 + 5
g (x) = 3x + 3 or g (x) = -3x + 7

c.
The given absolute value function is:
h (x) = 3 | x – 2 | + 5
So,
h (x) = 3 (x – 2) + 5 or h (x) = -3 (x – 2) + 5
h (x) = 3x – 6 + 5 or h (x) = -3x + 6 + 5
h (x) = 3x – 1 or h (x) = -3x + 11
Compare the solutions of the given absolute value functions with
f (x) = mx + c
Where,
m is the slope
c is the y-intercept
Hence, from the above,
We can conclude that a) and b ) have the same y-intercepts

Monitoring Progress and Modeling with Mathematics

In Exercises 5–12, graph the function. Compare the graph to the graph of f(x) = | x |. Describe the domain and range.

Question 5.
d(x) = | x | – 4
Answer:

Question 6.
r(x) = | x | + 5
Answer:
The given functions are:
f (x) = | x |
r (x) = | x | + 5
Hence,
The representation of f (x) and r (x) in the coordinate plane is:

Hence, from the above,
We can conclude that r (x) is 5 units away from f (x)
Now,
The domain of the given absolute value function is: -5 ≤ x ≤ 5
The range of the given absolute value function is: 5 ≤y ≤10

Question 7.
m(x) = | x + 1 |
Answer:

Question 8.
v(x) = | x – 3 |
Answer:
The given absolute value functions are:
f (x) = | x |
v (x) = | x – 3 |
Hence,
The representation of f(x) and v (x) in the coordinate plane is:

Hence, from the above,
We can conclude that v (x) translates 3 units away from f (x)
Now,
The domain of the given absolute value function is: -6 ≤ x ≤ 10
The range of the given absolute value function is: 6 ≤ y ≤ 10

Question 9.
p(x) = \(\frac{1}{3}\) | x |
Answer:

Question 10.
j(x) = 3 | x |
Answer:
The given absolute value functions are:
f (x) = | x |
j (x) = 3 | x |
Hence,
The representation of f (x) and j (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that j (x) vertically stretches 3 units away from f (x)
Now,
The domain of the given absolute value function is: -3.5 ≤ x ≤ 3.5
The range of the given absolute value function is: 0 ≤ y ≤ 10

Question 11.
a(x) = -5 | x |
Answer:

Question 12.
q(x) = – \(\frac{3}{2}\) | x |
Answer:
The given absolute value functions are:
f (x) = | x |
q (x) = –\(\frac{3}{2}\) | x |
Hence,
The representation of f (x) and q (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that q (x) shrinks by \(\frac{3}{2}\) units from f (x)
Now,
The domain of the given absolute value function is: -6.5 ≤ x ≤ 6.5
The range of the given absolute value function is: -10 ≤ y ≤ 0

In Exercises 13–16, graph the function. Compare the graph to the graph of f(x) = | x − 6 |.

Question 13.
h(x) = | x – 6 | + 2
Answer:

Question 14.
n(x) = \(\frac{1}{2}\) | x – 6 |
Answer:
The given functions are:
f (x) = | x – 6 |
n (x) = \(\frac{1}{2}\) | x – 6 |
Hence,
The representation of f (x) and n (x) in the coordinate plane is:

Hence, from the above,
We can conclude that n (x) vertically stretches \(\frac{1}{2}\) units of f (x)

Question 15.
k(x) = -3 | x – 6 |
Answer:

Question 16.
g(x) = | x – 1 |
Answer:
The given functions are:
f (x) = | x – 6 |
g (x) = | x – 1 |
Hence,
The representation of f (x) and g (x) in the coordinate plane is:

Hence, from the above,
We can conclude that g (x) translates 4 units away from f (x) on the positive x-axis

In Exercises 17 and 18, graph the function. Compare the graph to the graph of f(x) = | x + 3 | − 2.

Question 17.
y(x) = | x + 4 | – 2
Answer:

Question 18.
b(x) = | x + 3 | + 3
Answer:
The given absolute value functions are:
f (x) = | x + 3 | – 2
b (x) = | x + 3 | + 3
Hence,
The representation of f (x) and b (x) in the coordinate plane is:

Hence, from the above,
We can conclude that f (x) and b (x) are parallel lines since the slopes are constant and there are different values of the y-intercepts

In Exercises 19–22, compare the graphs. Find the value of h, k, or a.

Question 19.

Answer:

Question 20.

Answer:
The given graph is:

From the above graph,
The given functions are:
f (x) = | x |
t (x) = | x – h |
Where,
h is the horizontal translation
From the graph,
We can observe that the translation occurs at (1, 0) of f (x)
Hence, from the above,
We can conclude that the value of 'h' is: 1

Question 21.

Answer:

Question 22.

Answer:
The given graph is:

From the given graph,
The given graphs are:
f (x) = | x |
w (x) = a | x |
Where,
'a' is the vertical stretch
From the graph,
We can observe that the vertical stretch occurs at (0, 2)
Hence, from the above,
We can conclude that the value of a is: 2

In Exercises 23–26, write an equation that represents the given transformation(s) of the graph of g(x) = | x |.

Question 23.
vertical translation 7 units down
Answer:

Question 24.
horizontal translation 10 units left
Answer:
The given statement is:
Horizontal translation 10 units left
Let the absolute value function be y = | x |
So,
"Horizontal translation" means the moving of value on the x-axis either to the left side or the right side i.e., either on the negative or positive x-axis
So,
The given statement in the absolute value function form is:
y = | x | + 10

Question 25.
vertical shrink by a factor of \(\frac{1}{4}\)
Answer:

Question 26.
vertical stretch by a factor of 3 and a reflection in the x-axis
Answer:
The given statement is:
Vertical stretch by a factor of 3 and a reflection in the x-axis
We know that,
The representation of the vertical stretch 'a' of the function is:
y = a.f (x)
The representation of the reflection of the function is:
y= f (-x)
Now,
Let the absolute value function be
y = x
Hence,
The representation of the given statement in the absolute value function form is:
Vertical stretch:
y = 3x
Reflection:
y = -3x

In Exercises 27–32, graph and compare the two functions.

Question 27.
f(x) = | x – 4 |; g(x) = | 3x – 4 |
Answer:

Question 28.
h(x) = | x + 5 |; t(x) = | 2x + 5 |
Answer:
The given functions are:
h (x) = | x + 5 |
t (x) = | 2x + 5 |
Hence,
The representation of h (x) and t (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that t (x) translates 6 units away from h (x) in the negative y-axis

Question 29.
p(x) = | x + 1 | – 2; q(x) = | \(\frac{1}{4}\)x + 1 | – 2
Answer:

Question 30.
w(x) = | x – 3 | + 4; y(x) = | 5x – 3 | + 4
Answer:
The given absolute value functions are:
w (x) = | x – 3 | + 4
y (x) = | 5x – 3 | + 4
Hence,
The representation of w (x) and y (x) in the coordinate plane is:

Hence, from the above,
We can conclude that w (x) is 7 units away from y (x) in the positive y-axis

Question 31.
a(x) = | x + 2 | + 3; b(x) = | -4x + 2 | + 3
Answer:

Question 32.
u(x) = | x – 1 | + 2; v(x) = | –\(\frac{1}{2}\)x – 1 | + 2
Answer:
The given absolute value functions are:
u (x) = | x – 1 | + 2
v (x) = | –\(\frac{1}{2}\)x – 1 | + 2
Hence,
The representation of u (x) and v (x) in the coordinate plane is:

Hence, from the above,
We can conclude that u (x) is 1 unit away from v (x) in the positive y-axis

In Exercises 33–40, describe the transformations from the graph of f(x) = | x | to the graph of the given function. Then graph the given function.

Question 33.
r(x) = | x + 2 | – 6
Answer:

Question 34.
c(x) = | x + 4 | + 4
Answer:
The given absolute value function is:
c (x) = | x + 4 | + 4
Hence,
The representation of c (x) in the coordinate plane is:

Hence, from the above,
We can conclude that c (x) translates 4 units away in the positive x-axis and 4 units vertically in the y-axis

Question 35.
d(x) = – | x – 3 | + 5
Answer:

Question 36.
v(x) = -3| x + 1 | + 4
Answer:
The given absolute value function is:
v (x) = -3 | x + 1 | + 4
Hence,
The representation of v (x) in the coordinate plane is:

Hence, from the above,
We can conclude that v (x) translates 4 units vertically above and 2 units to the right side of the positive x-axis

Question 37.
m(x) = \(\frac{1}{2}\) | x + 4 | – 1
Answer:

Question 38.
s(x) = | 2x – 2 | – 3
Answer:
The given absolute value function is:
s (x) = | 2x – 2 | – 3
Hence,
The representation of s (x) in the coordinate plane is:

Hence, from the above,
We can conclude that s (x) translates 4 units away to the right side of the positive x-axis

Question 39.
j(x) = | -x + 1 | – 5
Answer:

Question 40.
n(x) = | –\(\frac{1}{3}\)x + 1 | + 2
Answer:
The given absolute value function is:
n (x) = | –\(\frac{1}{3}\)x + 1 | + 2
Hence,
The representation of n (x) in the coordinate plane is:

Hence, from the above,
We can conclude that n (x) translates 4 units away from the origin

Question 41.
MODELING WITH MATHEMATICS
The number of pairs of shoes sold s (in thousands) increases and then decreases as described by the function s(t) = -2 | t – 15 | + 50, where t is the time (in weeks).

a. Graph the function.
b. What is the greatest number of pairs of shoes sold in 1 week?
Answer:

Question 42.
MODELING WITH MATHEMATICS
On the pool table shown, you bank the five ball off the side represented by the x-axis. The path of the ball is described by the function p(x) = \(\frac{4}{3}\) | x – \(\frac{5}{4}\) |.

a. At what point does the five-ball bank off the side?
Answer:
It is given that on the pool table shown, you bank the five ball off the side represented by the x-axis. The path of the ball is described by the function
p(x) = \(\frac{4}{3}\) | x – \(\frac{5}{4}\) |
Now,
To find the point where the five-ball bank off the side, draw the plot of the given absolute value function
So,
The representation of the given absolute value function in the coordinate plane is:

Hence, from the above,
The point where the five-ball offside is: (1, 0)

b. Do you make the shot? Explain your reasoning.
Answer:
The given graph is:

From the graph,
We can observe that the point is accurately going through the hole i.e., at (5, 5)
Hence, from the above,
We can conclude that you make the shot

Question 43.
USING TRANSFORMATIONS
The points A (-\(\frac{1}{2}\), 3) , B(1, 0), and C(-4, -2) lie on the graph of the absolute value function f. Find the coordinates of the points corresponding to A, B, and C on the graph of each function.
a. g(x) = f(x) – 5
b. h(x) = f(x – 3)
c. j(x) = -f(x)
d. k(x) = 4f(x)
Answer:

Question 44.
USING STRUCTURE
Explain how the graph of each function compares to the graph of y = | x | for positive and negative values of k, h, and a.
a.y = | x | + k
b. y = | x – h |
c. y = a| x |
d. y = | ax |
Answer:
The given absolute value functions are:
a.y = | x | + k
b. y = | x – h |
c. y = a| x |
d. y = | ax |
We know that,
| x | = x for x > 0
| x | = -x for x < 0
It is given that
The parent function is:
y = | x |
Now,
a.
The given absolute value function is:
y = | x | + k
Hence,
The given absolute value function translates from the parent function k units away in the positive y-axis vertically
b.
The given absolute value function is:
y = | x – h |
Hence,
The given absolute value function translates from the parent function h units away in the positive x-axis horizontally
c.
The given absolute value function is:
y = a| x |
Hence,
The given absolute value function reflects from the parent function 'a' units away
d.
The given absolute value function is:
y = | ax |
Hence,
The given absolute value function dilates from the parent function with the value of 'a' units

ERROR ANALYSIS
In Exercises 45 and 46, describe and correct the error in graphing the function.

Question 45.

Answer:

Question 46.

Answer:
The given absolute value function is:
y = -3 | x|
We know that,
| x | = x for x > 0
| x | = -x for x < 0
So,
y = -3 (x) or y = -3 (-x)
So,
y = -3x or y = 3x
Hence,
The representation of the given absolute value function in the coordinate plane is:

Hence, from the above,
We can conclude that the given absolute value function in downward direction

MATHEMATICAL CONNECTIONS
In Exercises 47 and 48, write an absolute value function whose graph forms a square with the given graph.

Question 47.

Answer:

Question 48.

Answer:
The given graph is:

From the graph,
The given absolute value function is:
y = | x – 3 | + 1
From the given absolute value function,
We can say that the x-axis translates 3 units away in the positive x-axis and the y-axis is a vertical stretch with the value of k as 1

Question 49.
WRITING
Compare the graphs of p(x) = | x – 6 | and q(x) = | x | – 6.
Answer:

Question 50.
HOW DO YOU SEE IT? The object of a computer game is to break bricks by deflecting a ball toward them using a paddle. The graph shows the current path of the ball and the location of the last brick.

a. You can move the paddle up, down, left, and right. At what coordinates should you place the paddle to break the last brick? Assume the ball deflects at a right angle.
Answer:
The given graph is:

It is given that the paddle moves at the right angle using a ball
So,
From the graph,
The coordinates that are at a right angle to break the brick using a paddle is:
(3, 8) and (14, 8)

b. You move the paddle to the coordinates in part (a), and the ball is deflected. How can you write an absolute value function that describes the path of the ball?
Answer:
From part (a),
The coordinates that are used to break the brick using a paddle is:
(3, 8) and (14, 8)
We know that,
For absolute value function,
The parent function will always be:
f (x) = | x |
From part (a),
The y-axis is constant and the x-axis is translating 3 units away and 14 units away
We know that,
The value of the absolute value function with the translation of x-value and  y-value as k-value
So,
f (x) = | x – h | + k
Hence,
The absolute value function that describes the path of the ball is:
f (x) = | x – 3 | + 8
f (x) = | x – 14 | + 8

In Exercises 51–54, graph the function. Then rewrite the absolute value function as two linear functions, one that has the domain x < 0 and one that has the domain x ≥ 0.

Question 51.
y = | x |
Answer:

Question 52.
y = | x | – 3
Answer:
The given absolute value function is:
We know that,
| x | = x for x > 0
| x | = -x for x < 0
So,
y = x – 3 or y = -x – 3
Hence,
The representation of the given absolute value function in the coordinate plane is:

Question 53.
y = – | x | + 9
Answer:

Question 54.
y = – 4 | x |
Answer:
The given absolute value function is:
y = -4 | x |
We know that,
| x | = x for x > 0
| x | = -x for x < 0
So,
y = -4 (x) or y = -4 (-x)
y = -4x or y = 4x
Hence,
The representation of the given absolute value function in the coordinate plane is:

In Exercises 55–58, graph and compare the two functions.

Question 55.
f(x) = | x – 1 | + 2; g(x) = 4 | x – 1 | + 8
Answer:

Question 56.
s(x) = | 2x – 5 | – 6; t(x) = \(\frac{1}{2}\) | 2x – 5 | – 3
Answer:
The given functions are:
s (x) = | 2x – 5 | – 6
t (x) = \(\frac{1}{2}\) | 2x – 5 | – 3
Hence,
The representation of s (x) and t (x) in the coordinate plane is:

Hence, from the above,
We can conclude that s (x) translates 60 units away from t (x)

Question 57.
v(x) = -2 | 3x + 1 | + 4; w(x) = 3 | 3x + 1 | – 6
Answer:

Question 58.
c(x) = 4 | x + 3 | – 1; d(x) = –\(\frac{4}{3}\) | x + 3 | + \(\frac{1}{3}\)
Answer:
The given functions are:
c (x) = 4 | x + 3 | – 1
d (x) =-\(\frac{4}{3}\) | x + 3 | + \(\frac{1}{3}\)
Hence,
The representation of c (x) and d (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that d(x) and c (x) are opposite to each other

Question 59.
REASONING
Describe the transformations from the graph of g(x) = -2 | x + 1 | + 4 to the graph of h(x) =| x |. Explain your reasoning.
Answer:

Question 60.
THOUGHT-PROVOKING
Graph an absolute value function f that represents the route a wide receiver runs in a football game. Let the x-axis represent the distance (in yards) across the field horizontally. Let the y-axis represent the distance (in yards) down the field. Be sure to limit the domain so the route is realistic.
Answer:
It is given that
Let the x-axis represent the distance (in yards) across the field horizontally
Let the y-axis represent the distance (in yards) down the field.
Let,
The distance across the field horizontally is: 50 yards
The distance down the field is: 25 yards
Remember we can take any value in the x-axis and the y-axis
So,
(x, y) = (50, 25)
So,
The absolute value function f that represents the route a wide receiver runs in a football game is:
f (x) = | x – 50 | + 25
Hence,
The representation of the above absolute value function in the coordinate plane is:

Hence, from the above,
The domain of the given absolute value function is: x ≥ 0
The range of the given absolute value function is: y > 0

Question 61.
SOLVING BY GRAPHING
Graph y = 2 | x + 2 | – 6 and y = -2 in the same coordinate plane. Use the graph to solve the equation 2 | x + 2 | – 6 = -2. Check your solutions.
Answer:

Question 62.
MAKING AN ARGUMENT
Let p be a positive constant. Your friend says that because the graph of y =| x | + p is a positive vertical translation of the graph of y = | x |, the graph of y = | x + p | is a positive horizontal translation of the graph of y = | x |. Is your friend correct? Explain.
Answer:
It is given that your friend says that because the graph of y =| x | + p is a positive vertical translation of the graph of y = | x |, the graph of y = | x + p | is a positive horizontal translation of the graph of y = | x |.
Now,
The graph of
y = | x | + p is a positive vertical translation because the y-intercept in the given equation is positive
Now,
The graph of
y = | x + p | will be a horizontal translation with negative translation value
But, according to your friend,
y = | x + p | is a positive horizontal translation
Hence, from the above,
We can conclude that your friend is not correct

Question 63.
ABSTRACT REASONING
Write the vertex of the absolute value function f(x) = | ax – h | + k in terms of a, h, and k.
Answer:

Maintaining Mathematical Proficiency

Solve the inequality. (Section 2.4)

Question 64.
8a – 7 ≤ 2(3a – 1)
Answer:
The given inequality is:
8a – 7 ≤ 2 (3a – 1)
So,
8a – 7 ≤ 2 (3a) – 2 (1)
8a – 7 ≤ 6a – 2
8a – 6a ≤ -2 + 7
2a ≤ 5
a ≤ \(\frac{5}{2}\)
Hence, from the above,
We can conclude that the solution to the given inequality is:
a ≤ \(\frac{5}{2}\)

Question 65.
-3(2p + 4) > -6p – 5
Answer:

Question 66.
4(3h + 1.5) ≥ 6(2h – 2)
Answer:
The given inequality is:
4 (3h + 1.5) ≥ 6 (2h – 2)
So,
4 (3h) + 4 (1.5) ≥ 6 (2h) – 6 (2)
12h + 6 ≥ 12h – 12
12h – 12h + 6 ≥ -12
6 ≥ -12
Since the above inequality is true,
The given inequality has infinite solutions

Question 67.
-4(x + 6) < 2(2x – 9)
Answer:

Find the slope of the line. (Section 3.5)

Question 68.

Answer:
The given graph is:

From the above graph,
The points are: (0, 3), (-2, -2)
We know that,
The slope of the give line when (x1, y1), (x2, y2) are given is:
m= \(\frac{y2 – y1}{x2 – x1}\)
So,
(x1, y1) = (0, 3) and (x2, y2) = (-2, -2)
So,
m = \(\frac{-2 – 3}{-2 – 0}\)
m = \(\frac{-5}{-2}\)
m = \(\frac{5}{2}\)
Hence, from the above,
We can conclude that the slope of the given line is: \(\frac{5}{2}\)

Question 69

Answer:

Question 70.

Answer:
The given graph is:

From the above graph,
The given points are: (-3, 1), (1, -4)
We know that,
The slope of the give line when (x1, y1), (x2, y2) are given is:
m= \(\frac{y2 – y1}{x2 – x1}\)
So,
(x1, y1) = (-3, 1) and (x2, y2) = (1, -4)
So,
m = \(\frac{-4 – 1}{1 – [-3]}\)
m = \(\frac{-5}{1 + 3}\)
m = \(\frac{-5}{4}\)
Hence, from the above,
We can conclude that the slope of the given line is: –\(\frac{5}{4}\)

Graphing Linear Functions Performance Task: The Cost of a T-Shirt

3.4–3.7 What Did You Learn?

Core Vocabulary

Core Concepts
Section 3.4

Section 3.5

Section 3.6

Section 3.7

Mathematical Practices

Question 1.
Explain how you determined what units of measure to use for the horizontal and vertical axes in Exercise 37 on page 142.
Answer:
In Exercise 37 on page 142,
The given function is:
d (t)= (1/2)t + 6
We know that,
The horizontal axis represents the independent variable
The vertical axis represents the dependent variable
So,
The horizontal axis represents the time
The vertical axis represents the depth

Question 2.
Explain your plan for solving Exercise 48 on page 153.
Answer:
In Exercise 48 on page 153,
The profits of a school obtained by a school before playoffs and during the playoffs
So,
First, plot the 2 equations in a coordinate plane and compare the functions in the coordinate plane

Performance Task

The Cost of a T-Shirt

You receive bids for making T-shirts for your class fundraiser from four companies. To present the pricing information, one company uses a table, one company uses a written description, one company uses an equation, and one company uses a graph. How will you compare the different representations and make the final choice? To explore the answers to this question and more, go to

Graphing Linear Functions Chapter Review

3.1 Functions (pp. 103 – 110)

Determine whether the relation is a function. Explain.

Answer:
The given table is:

We know that,
Every input has exactly one output.
x represents the input
y represents the output
Hence, from the table,
We can conclude that the given table is a function

Determine whether the relation is a function. Explain.

Question 1.
(0, 1), (5, 6), (7, 9)
Answer:
The given ordered pairs are:
(0, 1), (5, 6), (7, 9)
We know that,
Every input has exactly one output
x represents the input
y represents the output
Hence, from the above,
We can conclude that the given relation is a function

Question 2.

Answer:
The given graph is:

We know that,
Every input has exactly one output
x represents the input
y represents the output
Now,
From the graph,
The input '2' has 2 outputs i.e., (2, 0), and (2, 2)
Hence, from the above,
We can conclude that the given graph is not a function

Question 3.

Answer:
The given relation is:

We know that,
Every input has exactly one output
x represents the input
y represents the output
Hence, from the above,
We can conclude that the given relation is a function

Question 4.
The function y = 10x + 100 represents the amount y (in dollars) of money in your bank account after you babysit for x hours.
a. Identify the independent and dependent variables.
Answer:
The given function is:
y = 10x + 100
Where,
y represents the amount in dollars
x represents the number of hours
Now,
From the given function,
WE can say that
The independent variable of the given function is: x
The dependent variable of the given function is: y

b. You babysit for 4 hours. Find the domain and range of the function.
Answer:
The given function is:
y = 10x + 100
Where,
y represents the amount in dollars
x represents the number of hours
It is given that you babysit for 4 hours
So,
The given value of x is: 4
So,
The maximum value of y is:
y = 10 (4) + 100
y = 40 + 100
y = $140
We know that,
The amount and the time must not be the negative values
Hence,
The domain of the given function is: 0 ≤ x ≤ 4
The range of the give function is: 0 ≤ y ≤ 140

3.2 Linear Functions (pp. 111–120)

Does the table or equation represent a linear or nonlinear function? Explain.

Answer:
As x increases by 4, y increases by different amounts. The rate of change is not constant.
So,
The function is nonlinear.

b. y = 3x – 4
Answer:
The equation is in the form y = mx + b.
So,
The equation represents a linear function.

Does the table or graph represent a linear or nonlinear function? Explain.

Question 5.

Answer:
The given table is:

From the given table,
The difference between all the values of x is: 5
The difference between all the values of y is: -3
Since there is a constant difference present between the values of x and the values of y,
The given table is a linear function

Question 6.

Answer:
The given graph is:

We know that,
The representation of a linear function in the graph is a "Straight line"
But,
We can say that the given graph is not a straight line from the graph
Hence, from the above,
We can conclude that the given graph is a non-linear function

Question 7.
The function y = 60 – 8x represents the amount y (in dollars) of money you have after buying x movie tickets.
(a) Find the domain of the function. Is the domain discrete or continuous? Explain.
Answer:
The given function is:
y = 60 – 8x
Where,
y represents the amount in dollars
x represents the number of movie tickets
We know that,
The domain is the set of all the values of x so that the given function will be satisfied
Since x represents the number of movie tickets, the value of x can't be negative and the number of tickets will be infinity
Hence,
The domain of the given function is: 0 ≤ x ≤ ∞
Hence, from the above domain,
We can conclude that the domain is continuous

(b) Graph the function using its domain. Evaluate the function when x = -3, 0 and 5.
Answer:
The given function is:
y = 60 – 8x
Now,
The value of the given function when x = -3 is:
y = 60 – 8 (-3)
y = 60 + 24
y = 84
The value of the given function when x = 0 is:
y = 60 – 8 (0)
y = 60
The value of the given function when x = 5 is:
y = 60 – 8(5)
y = 60 – 40
y = 20
Now,
The representation of the given function along with its domain is:

Question 8.
f(x) = x + 8
Answer:
The given equation is:
f (x) = x + 8
We know that,
The standard representation of the function f (x) is: y
The standard representation of the linear equation is:
y = mx + c
Now,
y = x + 8
By comparing the given equation and the standard representation of the linear equation,
We can conclude that the given equation is a linear equation

Find the value of x, so that the function has the given value.

Question 9.
g(x) = 4 – 3x

Question 10.
k(x) = 7x; k(x) = 49
Answer:
The given function is:
k (x) = 7x with k (x) = 49
So,
49 = 7x
x = 49 / 7
x = 7
Hence, from the above,
We can conclude that the value of the given function is: 7

Question 11.
r(x) = -5x – 1; r(x) = 19
Answer:
The given function is:
r (x) = 5x – 1 with r (x) = 19
So,
19 = 5x – 1
5x = 19 + 1
5x = 20
x = 20 / 5
x = 4
Hence, from the above,
We can conclude that the value of the given function is: 4

Graph the linear function.

Question 12.
g(x) = -2x – 3
Answer:
The given function is:
g (x) = -2x – 3
Hence,
The representation of g (x) in the coordinate plane is:

Question 13.
h(x) = \(\frac{2}{3}\)x + 4
Answer:
The given function is:
h (x) = \(\frac{2}{3}\)x + 4
Hence,
The representation of h (x) in the coordinate plane is:

Question 14.
8x – 4y = 16
Answer:
The given function is:
8x – 4y = 16
4y = 8x – 16
y = \(\frac{8x  16}{4}\)
y = \(\frac{8x}{4}\) – \(\frac{16}{4}\)
y =2x – 4
Hence,
The representation of the given function in the coordinate plane is:

Question 15.
-12x – 3y = 36
Answer:
The given function is:
-12x – 3y = 36
Hence,
The representation of the given function in the coordinate plane is:

Question 16.
y = -5
Answer:
The given function is:
y = -5
Hence,
The representation of the given function in the coordinate plane is:

Question 17.
x = 6
Answer:
The given function is:
x = 6
Hence,
The representation of the given function in the coordinate plane is:

The points represented by the table lie on a line. Find the slope of the line.

Question 18.

Answer:
The given table is:

We know that,
If the input and output values are given in the form of the table,
We can take any 2 pairs of the input and output values to find the slope
So,
The representation of ordered pairs to find the slope is:
(x1, y1) = (6, 9),
(x2, y2) = (11, 15)
Now,
We know that,
The slope of the line = \(\frac{y2 – y1}{x2 – x1}\)
= \(\frac{15 – 9}{11 – 6}\)
= \(\frac{6}{5}\)
Hence, from the above,
We can conclude that the slope of the given line is: \(\frac{6}{5}\)

Question 19.

Answer:

The given table is:

We know that,
If the input and output values are given in the form of the table,
We can take any 2 pairs of the input and output values to find the slope
So,
The representation of ordered pairs to find the slope is:
(x1, y1) = (3, -5),
(x2, y2) = (3, -2)
Now,
We know that,
The slope of the line = \(\frac{y2 – y1}{x2 – x1}\)
= \(\frac{-2 – [-5]}{3 – 3}\)
= \(\frac{5 – 2}{0}\)
= Undefined or ∞
Hence, from the above,
We can conclude that the slope of the given line is: Undefined or ∞

Question 20.

Answer:
The given table is:

We know that,
If the input and output values are given in the form of the table,
We can take any 2 pairs of the input and output values to find the slope
So,
The representation of ordered pairs to find the slope is:
(x1, y1) = (-4, -1),
(x2, y2) = (-3, -1)
Now,
We know that,
The slope of the line = \(\frac{y2 – y1}{x2 – x1}\)
= \(\frac{-1 – [-1]}{-3 – [-4]}\)
= \(\frac{-1 + 1}{-3 + 4}\)
= 0
Hence, from the above,
We can conclude that the slope of the given line is: 0

Graph the linear equation. Identify the x-intercept.

Question 21.
y = 2x + 4
Answer:
The given linear equation is:
y = 2x + 4
To find the x-intercept, put y = 0
So,
2x + 4 = 0
2x = -4
x = -4 / 2
x = -2
Hence,
The representation of the given linear equation in the coordinate plane is:

Question 22.
-5x + y = -10
Answer:
The given linear equation is:
-5x + y = -10
To find the x-intercept, put y = 0
-5x + 0 = -10
-5x = -10
5x = 10
x = 10 / 5
x = 2
Hence,
The representation of the given linear equation in the coordinate plane is:

Question 23.
x + 3y = 9
Answer:
The given linear equation is:
x + 3y = 9
To find the x-intercept, put y = 0
So,
x + 0 = 9
x = 9
Hence,
The representation of the given linear equation in the coordinate plane is:

Question 24.
A linear function h models a relationship in which the dependent variable decreases 2 units for every 3 units the independent variable increases. Graph h when h(0) = 2. Identify the slope, y-intercept, and x-intercept of the graph. Let f(x) = 3x + 4. Graph f and h. Describe the transformation from the graph of f to the graph of h.
Answer:
It is given that a linear function h models a relationship in which the dependent variable decreases 2 units for every 3 units the independent variable increases.
We know that,
x represents the independent variable
y represents the dependent variable
So,
The given x value is: 3
The given y value is: -2
So,
The rate of change (or) slope (m) = \(\frac{y}{x}\)
= \(\frac{-2}{3}\)
= –\(\frac{2}{3}\)
It is also given that h (0) = 2
That means the y-intercept of h (x) is 2 when x is 0
We know that,
The representation of the standad form of the linear equation is:
y = mx + c
So,
h (x) =-\(\frac{2}{3}\)x + 2
The other given function is:
f (x) = 3x + 4
Hence,
The representation of f (x0 and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that h (x) is 1.5 units away from f (x) on the x-axis

Let f(x) = 3x + 4.Graph f and h. Describe the transformation from the graph of f to the graph of h.

Question 25.
h(x) = f(x + 3)
ANswer:
The given functions are:
f (x) = 3x + 4
h (x) = f (x + 3)
So,
h (x) = 3 (x + 3) + 4
h (x) = 3 (x) + 3 (3) + 4
h (x) = 3x + 9 + 4
h (x) = 3x + 13
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that f (x) and h (x) are parallel lines

Question 26.
h(x) = f(x) + 1
Answer:
The given functions are:
f (x) = 3x + 4
h (x) = f (x) + 1
So,
h (x) = 3x + 4 + 1
h (x) = 3x + 5
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that f (x) and h (x) are the paralel lines

Question 27.
h(x) = f(-x)
Answer:
The given functions are:
f (x) = 3x + 4
h (x) = f (-x)
So,
h (x) = 3 (-x) + 4
h (x) = -3x + 4
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that h (x) and f (x) are inversions of each other

Question 28.
h(x) = -f(x)
Answer:
The given functions are:
f (x) = 3x + 4
h (x) = -f (x)
So,
h (x) = – (3x + 4)
h (x) = -3x – 4
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that h (x) and f (x) are reflections to each other

Question 29.
h(x) = 3f(x)
Answer:
The given function srae:
f (x) = 3x + 4
h (x) = 3 f (x)
So,
h (x) = 3 (3x + 4)
Hence,
The representation of f (x) and  h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that h (x) is a vertical stretch of f (x)

Question 30.
h(x) = f(6x)
Answer:
The given functions are:
f (x) = 3x + 4
h (x) = f (6x)
So,
h (x) = 3 (6x) + 4
h (x) = 18x + 4
Hence,
The representation of f (x) and h (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that h (x) translates 2 units away from f (x) on the x-axis

Question 31.
Graph f(x) = x and g(x) = 5x + 1. Describe the transformations from the graph of f to the graph of g.
Answer:
The given functions are:
f (x) = x
g (x) = 5x + 1
Hence,
The representation of f (x) and g (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that g (x) translates away 8 units away from f (x)  on the x-axis

Graph the function. Compare the graph to the graph of f(x) = | x |. Describe the domain and range.

Question 32.
m(x) = | x | + 6
Answer:
The given absolute value functions are:
f (x) = | x |
m (x) = | x | + 6
Hence,
The representation of f (x) and m (x) in the same coordinate plane is:

Hene, from the above,
We can conclude that m (x) translates 6 units up from the positive x-axis
Hence,
The domain of f (x) is: -10 ≤ x ≤ 10
The domain of m (x) is: -4 ≤ x ≤ 4
The range of m (x) is: 0 ≤y ≤ 10
The range of f (x) is: 0 ≤ y ≤10
The range of m (x) is: 6 ≤y ≤10

Question 33.
p(x) = | x – 4 |
Answer:
The given absolute value functions are:
f (x) = | x |
p (x) = | x – 4 |
Hence,
The representation of f (x) and p (x) in the same coordinate plane is:

Hene, from the above,
We can conclude that p (x) translates 1 unit up from the positive x-axis
Hence,
The domain of f (x) is: -10 ≤ x ≤ 10
The domain of p (x) is: -6 ≤ x ≤ 10
The range of f (x) is: 0 ≤ y ≤10
The range of p (x) is: 6 ≤y ≤10

Question 34.
q(x) = 4 | x |
Answer:
The given absolute value functions are:
f (x) = | x |
q (x) = 4 | x |
Hence,
The representation of f (x) and q (x) in the same coordinate plane is:

Hene, from the above,
We can conclude that q (x) translates 7 units up from the positive x-axis
Hence,
The domain of f (x) is: -10 ≤ x ≤ 10
The domain of q (x) is: -2.5 ≤ x ≤ 2.5
The range of q (x) is: 0 ≤y ≤ 10
The range of f (x) is: 0 ≤ y ≤10

Question 35.
r(x) = –\(-\frac{1}{4}\)| x |
Answer:
The given absolute value functions are:
f (x) = | x |
r (x) = –\(-\frac{1}{4}\) | x |
So,
r (x) = \(\frac{1}{4}\) | x |
Hence,
The representation of f (x) and r (x) in the same coordinate plane is:

Hene, from the above,
We can conclude that r (x) translates  units up from the positive x-axis
Hence,
The domain of f (x) is: -10 ≤ x ≤ 10
The domain of r (x) is: -10 ≤ x ≤ 10
The range of r (x) is: 0 ≤y ≤ 2
The range of f (x) is: 0 ≤ y ≤10

Question 36.
Graph f(x) = | x – 2 | + 4 and g(x) = | 3x – 2 | + 4. Compare the graph of g to the graph of f.
Answer:
The given functions are:
f (x) = | x – 2 | + 4
g (x) = | 3x – 2 | + 4
Hence,
The representation of f (x) and g (x) in the same coordinate plane is:

Hence, from the above,
We can conclude that g (x) translates 5 units away from f (x) on the x-axis

Question 37.
Let g(x) = \(\frac{1}{3}\) | x – 1 | – 2.
(a) Describe the transformations from the graph of f(x) = | x | to the graph of g.
Answer:
The given functions are:
f (x) = | x |
g (x) = \(\frac{1}{3}\) | x – 1 | – 2
Hence,
The representation of f (x) and g (x) in the coordinate plane is:

Hence, from the above,
We can conclude that g (x) translates 2 units away from f (x) on the y-axis
(b) Graph g.
Answer:
It is given that
g (x) = \(\frac{1}{3}\) | x – 1 | – 2
Hence,
The representation of g (x) in the coordinate plane is:

Graphing Linear Functions Chapter Test

Determine whether the relation is a function. If the relation is a function, determine whether the function is linear or nonlinear. Explain.

Question 1.

Answer:
The given table is:

We know that,
Every input has exactly one output
So,
The given table is a function
Now,
The constant between all the values of x is: 1
There is no constant difference between all the values of y
Since there is no constant rate of change,
We can conclude that the obtained function is a non-linear function

Question 2.
y = -2x + 3
Answer:
The given equation is:
y = -2x + 3
We know that,
The standard form of the linear equation is:
y = mx + c
For a relation o be a function, every input has exactly only one output
Hence, from the above,
We can conclude that the given equation is a function and it is a linear function

Question 3.
x = -2
Answer:
The given equation is:
x = -2
We know that,
A relation is said to be a function if every input has exactly one output
So,
For x = -2
The input is: 2
The output is: 0
So,
The given equation is a function
For a given function to be a linear function, it will be in the form
y = mx + c
Hence, from the above,
We can conclude that the given function is a non-linear function

Graph the equation and identify the intercept(s). If the equation is linear, find the slope of the line.

Question 4.
2x – 3y = 6
Answer:
The given equation is:
2x – 3y = 6
3y = 2x – 6
y = \(\frac{2x – 6}{3}\)
y = \(\frac{2x}{3}\) – \(\frac{6}{3}\)
y = \(\frac{2}{3}\)x – 2
Hence,
The above equation is in the form of
y = mx + c
So,
The given equation is a linear equation
So,
m = \(\frac{2}{3}\)
The y-intercept is: -2
To find the x-intercept, put y = 0
\(\frac{2}{3}\)x – 2 = 0
x = 3
Hence, from the above,
We can conclude that
Slope (m) = \(\frac{2}{3}\)
The x-intercept is: 3
The y-intercept is: -2

Question 5.
y = 4.5
Answer:
The given equation is:
y = 4.5
The given equation is not in the form of
y = mx + c
Hence,
The given equation is non-linear

Question 6.
y = | x – 1 | – 2
Answer:
The given equation is:
y = | x – 1 | – 2
We know that,
| x | = x for x > 0
| x | = -x for x < 0
So,
y = x – 1 – 2 or y = -(x – 1) – 2
y = x – 3 or y = -x + 1 – 2
y = x – 3 or y = -x – 1
The above 2 equations are in the form of
y = mx + c
Hence,
The slope is: 1 or -1
The y-intercept is: -3 or -1
To find the x-intercept, put y =0
So,
0 = x – 3 or 0 = -x – 1
x = 3 or x = -1
Hence, from the above,
We can conclude that
The slope is: 1 or -1
The x-intercept is: 3 or -1
The y-intercept is: -3 or -1

Find the domain and range of the function represented by the graph. Determine whether the domain is discrete or continuous. Explain.

Question 7.

Answer:
The given graph is:

From the graph,
We can observe that the graph is not a straight line
So,
We can say that the given graph is a non-linear function
Hence, from the above,
We can conclude that the domain is continuous by observing the graph

Question 8.

Answer:
The given graph is:

From the graph,
We can observe that the points form a straight line
So,
We can say that the given graph is a linear function
Hence, from the above,
We can conclude that the domain is continuous since all the points are connected

Graph f and g. Describe the transformations from the graph of f to the graph of g.

Question 9.
f(x) = x; g(x) = -x + 3
Answer:
The given functions are:
f (x) = x
g (x) = -x + 3
Hence,
The representation of f (x) and g (x) in the coordinate plane is:

Hence, from the above,
We can conclude that f (x) and g (x) are perpendicular lines with a slope of -1

Question 10.
f(x) = | x | ; g(x) = | 2x + 4 |
Answer:
The given functions are:
f (x) = | x |
g (x) = | 2x + 4 |
Hence,
The representation of f (x) and g (x) in the coordinate plane is:

Hence, from the above,
We can conclude that g (x) is 2 units away from f (x) on the x-axis

Question 11.
Function A represents the amount of money in a jar based on the number of quarters in the jar. Function B represents your distance from home over time. Compare the domains.
Answer:
It is given that
Function A represents the amount of money in a jar based on the number of quarters in the jar
Function B represents your distance from home over time
We know that,
The distance should be greater than or equal to 0
The amount is greater than or equal to a quarter of the amount of the money
Hence,
The domain of function A is: \(\frac{1}{4}\) ≤ x ≤ ∞
The domain of function B is: 0 ≤ x ≤ ∞

Question 12.
A mountain climber is scaling a 500-foot cliff. The graph shows the elevation of the climber over time.
a. Find and interpret the slope and the y-intercept of the graph.
b. Explain two ways to find f(3). Then find f(3) and interpret its meaning.
c. How long does it take the climber to reach the top of the cliff? Justify your answer.

Answer:
a.
It is given that a mountain-climber is scaling a 500-foot cliff
The given graph is:

From the graph,
The equation that shows the elevation of the climber over tie is:
f (x) = 125x + 50
We know that,
The standard representation of the output for the function output is: y
So,
y = 125x + 50
Compare the above equation with
y = mx + c
So,
m = 125 and the y-intercept is: 50
Hence, from the above,
We can conclude that
The slope of the given equation is: 125
The y-intercept of the given equation is: 50

b.
From part (a),
The given equation is:
f (x) = 125x + 50
So,
f (3) = 125 (3) + 50
f (3) = 375 + 50
f (3) = 425
Hence, from the above,
We can conclude that for 3 hours, the climber climbs 425 feet

c.
The given graph is:

From the given graph,
The top of the hill is the maximum height i.e., the highest value on the y-axis
Hence, from the above,
We can conclude that it takes 4 hours to reach the top of the cliff

Question 13.
Without graphing, compare the slopes and the intercepts of the graphs of the functions f(x) = x + 1 and g(x) = f(2x).
Answer:
The given functons are:
f (x) = x + 1
g (x) = f (2x)
So,
g (x) = 2x + 1
Now,
Compare f (x) and g (x) with the standard linear equation
y = mx + c
So,
For f (x),
m = 1 and c = 1
Where,
c is the y-intercept
For g (x),
m = 2 and c = 1
Where,
c is the y-intercept

Question 14.
A rock band releases a new single. Weekly sales s (in thousands of dollars) increase and then decrease as described by the function s(t) = -2 | t – 20 | + 40, where t is the time (in weeks).
a. Identify the independent and dependent variables.
b. Graph s. Describe the transformations from the graph of f(x) = | x | to the graph of s.
Answer:
a.
It is given that a rock band releases a new single and weekly sales s (in thousands of dollars) increase and then decrease as described by the function
s(t) = -2 | t – 20 | + 40
where,
t is the time (in weeks)
Now,
The independent variable of the given function  is: t
The dependent variable of the given function is: s (t)

b.
The given absolute value functions are:
f (x) = | x |
s (t) = -2 | t – 20 | + 40
Hence,
The representation of f (x) and s (t) in the coordinate plane is:

Hence, from the above,
We can conclude that s (t) translates 5 units away from f (x) on the y-axis

Graphing Linear Functions Cumulative Assessment

Question 1.
You claim you can create a table of values that represents a linear function. Your friend claims he can create a table of values that represents a nonlinear function. Using the given numbers, what values can you use for x (the input) and y (the output) to support your claim? What values can your friend use?

Answer:
It is given that you claim you can create a table of values that represents a linear function. Your friend claims he can create a table of values that represents a nonlinear function.
Hence,
The values you and your friend use are:

Question 2.
A car rental company charges an initial fee of $42 and a daily fee of $12.
a. Use the numbers and symbols to write a function that represents this situation.

b. The bill is $138. How many days did you rent the car?
Answer:
a.
It is given that a car rental company charges an initial fee of $42 and a daily fee of $12
So,
The function that represents the situation is:
f (x) = (42 + 12) × x
f (x) = 52x
Where,
x is the number of days that you rent the car

b.
It is given that the bill is: $138
So,
138 = 52x
x = 138 ÷ 52
x = 3
Hence, from the above,
We can conclude that you can rent the car for 3 days

Question 3.
Fill in values for a and b so that each statement is true for the inequality ax − b> 0.
a. When a = _____ and b = _____, x > \(\frac{b}{a}\).
b. When a = _____ and b = _____, x < \(\frac{b}{a}\).
Answer:
a.
The given inequality is:
ax – b > 0
ax > b
x > b / a
So,
The value of a is less than b and the value of b is greater than a
b.
The value of a is greater than b and the value of b is less than a

Question 4.
Fill in the inequality with <, ≤, >, or ≥ so that the solution of the inequality is represented by the graph.

Answer:
The given number line is:

From the given number line,
The given inequality is:
-3 (x + 7) _____ -24
-3 (x) – 3 (7) _____ -24
-3x – 21 _____ -24
-3x ____ -24 + 21
-3x ____-3
3x ____ 3
x ____ 3 / 3
x ____ 1
From the number line
The marked line represented from 1 including 1 and continued till the right end of the number line
Hence,
x ≥ 1
Hence, from the above,
We can conclude that the symbol used for the given inequality is: ≥

Question 5.
Use the numbers to fill in the coefficients of ax + by = 40 so that when you graph the function, the x-intercept is -10 and the y-intercept is 8.

Answer:
The given function is:
ax + by = 40
It is given that the x-intercept and the y-intercept is: -10 and 8
Now,
To find the x-intercept, put y = 0
So,
ax = 40
a (-10) = 40
a = -40 / 10
a = -4
To find the y-intercept, put x = 0
So,
by = 40
b (8) = 40
b = 40 / 8
b = 5
Hence, from the above,
We can conclude that the values of a and b are: -4 and 5

Question 6.
Solve each equation. Then classify each equation based on the solution. Explain your reasoning.
a. 2x – 9 = 5x – 33
Answer:
The given expression is:
2x – 9 = 5x – 33
2x – 5x = -33 + 9
-3x = -24
x = 24 / 3
x = 8
Hence, from the above,
We can conclude that the given expression has 1 solution

b. 5x – 6 = 10x + 10
Answer:
The given expression is:
5x – 6 = 10x + 10
5x – 10x = 10 + 6
-5x = 16
x = -16 / 5
Hence, from the above,
We can conclude that the given expression has only 1 solution

c. 2(8x – 3) = 4(4x + 7)
Answer:
The given expression is:
2 (8x – 3 ) = 4 (4x + 7)
2 (8x) – 2 (3) = 4 (4x) + 4(7)
16x – 6 = 16x + 28
Hence, from the above,
We can conclude that the given expression has no solution

d. -7x + 5 = 2(x – 10.1)
Answer:
The given expression is:
-7x + 5 = 2 (x – 10.1)
-7x + 5 = 2x – 20.2
-7x – 2x = -20.2 – 5
-9x = -25.2
x = 25.9 / 9
Hence, from the above,
We can conclude that the given expression has only 1 solution

e. 6(2x + 4) = 4(4x + 10)
Answer:
The given expression is:
6 (2x + 4) = 4 (4x + 10)
12x + 24 = 16x + 40
12x – 16x = 40 – 24
-4x = 16
x = -16 / 4
x = -4
Hence, from the above,
We can conclude that the given expression has only 1 solution

f. 8(3x + 4) = 2(12x + 16)
Answer:
The given expression is:
8 (3x + 4) = 2 (12x + 16)
24x + 32 = 24x + 32
Hence, from the above,
We can conclude that the given expression has no solution

Question 7.
The table shows the cost of bologna at a deli. Plot the points represented by the table in a coordinate plane. Decide whether you should connect the points with a line. Explain your reasoning.

Answer:
The given table is:

Hence,
The representation of the points in the coordinate plane is:

Hence, from the above,
We can say that we can connect the points in the graph

Question 8.
The graph of g is a horizontal translation right, then a vertical stretch, then a vertical translation down of the graph of f(x) = x. Use the numbers and symbols to create g.

Answer:
The given function is:
f (x) = x
We know that,
f (x) can be re-written as f
g (x) can b ere-written as g
Now,
Let the operations performed on f (x) can be expressed in terms of g (x)
So,
Horizontal translation right:
g (x) = f (x) – 3 or g (x) = f (x) – 1 or g (x)  f (x) – (1/2)
Vertical stretch:
g (x) = 3 f (x)
Vertical translation down:
g (x) = f (x – 1)

Question 9.
What is the sum of the integer solutions of the compound inequality 2 | x – 5 | < 16?
A. 72
B. 75
C. 85
D. 88
Answer:
The given compound inequality is:
2 | x – 5 | < 16
So,
2 (x – 5) < 16 an d 2 (x – 5 ) > -16
2x – 10 < 16 and 2x – 10 > -16
2x < 26 and 2x >-6
x < 13 and x > -3
Hence,
The solution of the given compoud inequality is: -3 < x < 13
Now,
The sum of all the integers = -2 – 1 + 0 + 1 + 2 + 3 + 4 + 5+ 6 + 7 + 8 +9 + 10 + 11 + 12
= 75
Hence, from the above,
We can conclude that the sum of all the integers is: 75

Question 10.
Your bank offers a text alert service that notifies you when your checking account balance drops below a specific amount. You set it up so you are notified when your balance drops below $700. The balance is currently $3000. You only use your account for paying your rent (no other deposits or deductions occur). Your rent each month is $625.
a. Write an inequality that represents the number of months m you can pay your rent without receiving a text alert.
Answer:
It is given that your bank offers a text alert service that notifies you when your checking account balance drops below a specific amount. You set it up so you are notified when your balance drops below $700. The balance is currently $3000. You only use your account for paying your rent (no other deposits or deductions occur). Your rent each month is $625
Let m be the number of months
Hence,
The inequality that represents the number of months you can pay rent without receiving a text is:
3000 – 625x > 700

b. What is the maximum number of months you can pay your rent without receiving a text alert?
Answer:
From part (a),
The inequality that represents the number of months m without receiving a text alert is:
3000 – 625x > 700
3000 – 700 > 625x
2300 > 625x
2300 / 625 > x
3.68 > x
x < 4 months [ Since the number of months will not be in decimals]
Hence, from the above,
We can conclude that the maximum number of months you can pay your rent without receiving a text alert is: 4 months

c. Suppose you start paying rent in June. Select all the months you can pay your rent without making a deposit.

Answer:
From part (b),
The maximum number of months is:
x < 4 months
It is given that you start paying rent in June
So,
All the months you can pay your rent without making a deposit is:
June, July, August

Monitoring Progress And Modeling With Mathematics 3.2 Answers

Source: https://ccssmathanswers.com/big-ideas-math-algebra-1-answers-chapter-3/

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