(8) Chapter 4 Study Guide

To
• represent linear relationships using
words, equations, tables, and graphs
Course 3, Lesson 4-1
Functions

1
Need Another Example?
2
Step-by-Step Example
1.
Write an equation to find the number
of liters in any number of quarts.
Describe the relationship in words.
Let ℓ represent the liters and q represent the quarts.
The equation is ℓ = 0.95q.
The table shows the number of liters in quarts of liquid.
The rate of change is the rate that
describes how one quantity changes
in relation to another quantity. The
rate of change of quarts to liters
is = or 0.95 liter in
every quart.

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2.
There are about 7.6 liters in 8 quarts.
About how many liters are in 8 quarts?
ℓ = 0.95q Write the equation.
ℓ = 0.95(8) Replace q with 8.
ℓ = 7.6 Multiply.

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5
3.
To find the y-intercept, use the slope and the coordinates of a
point to write the equation of the line in slope-intercept form.
The total distance Marlon ran in one
week is shown in the graph.
Write an equation to find the number of
miles run y after any number of days x.
Find the rate of change or the slope of the line.
Definition of slope
(x1, y1) = (2, 7); (x2, y2) = (4, 14)
Simplify.
Slope-intercept formy = mx + b
Replace m with the slope, 3.5.y = 3.5x + b
Use the point (2, 7). x = 2, y = 77 = 3.5(2) + b
Solve for b.0 = b
The slope is 3.5 and the y-intercept is 0. So, the
equation of the line is y = 3.5x + 0 or y = 3.5x.

Course 3, Lesson 4-
1
Functions
Words Distance traveled is equal to 12 miles per second times the
number of seconds.
Equation d = 12s
Table Graph

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6. Chloe competes in jump rope competitions. Her average
rate is 225 jumps per minute.
Make a table to find the number of jumps in 1, 2, 3, 4, or
5 minutes. Then graph the ordered pairs.

To
• express a relation as a table and a graph,
• determine the domain and range for each
relation
Course 3, Lesson 4-
2
Functions

Course 3, Lesson 4-
2
Functions
Ordered Pairs Table Graph

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2
3
1. Express the relation {(2, 6), (–4, –8), (–3, 6), (0, –4)} as
a table and a graph. Then state the domain and range.
Place the ordered
pairs in a table with
x-coordinates in the
first column and the
y-coordinates in the
second column.
Graph the
ordered pairs
on a coordinate
plane.
The domain is {–4, –3, 0, 2}. The range is {–8, –4, 6}.

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2. It costs $3 per hour to park at the Wild Wood
Amusement Park.
Make a table of ordered pairs in which the x-coordinate
represents the hours and the y-coordinate represents
the total cost for 3, 4, 5, and 6 hours.
a.
b. Graph the ordered pairs.

To
• find the value of a function for a
certain number,
• make a function table of values
Course 3, Lesson
4-3
Functions

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2
3
4
1. Find f(–3) if f(x) = 2x + 1.
f(x) = 2x + 1 Write the function.
f(–3) = 2(–3) + 1 Substitute –3 for x into the function rule.
f(–3) = –6 + 1 or –5 Simplify.
So, f(–3) = –5.

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2. Choose four values for x to make a function table
for f(x) = x + 5. Then state the domain and range of
the function.
Substitute each domain value x into the function rule.
Then simplify to find the range value.
The domain is {–2, –1, 0, 1}.
The range is {3, 4, 5, 6}.

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5
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5. There are approximately 770 peanuts in a jar of peanut butter.
The total number of peanuts p(j) is a function of the number
of jars of peanut butter j.
The function p(j) = 770j represents the situation.
Write a function to represent the total number of peanuts. Then
determine the number of peanuts in 7 jars of peanut butter.
Words The number
of peanuts equals
770
times
the number
of jars
p(j) = 770 • jFunction
To find the number of peanuts in 7 jars of peanut butter,
substitute 7 for j.
p(j) = 770j Write the function.
p(j) = 770(7) or 5,390 Substitute 7 for j.
There are 5,390 peanuts in 7 jars of peanut butter.

To represent a function by
• writing an equation,
• completing a table,
• graphing the function on the
coordinate plane
Course 3, Lesson
4-4
Functions

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2
3
1. The school store sells book covers for $2 each and notebooks
for $1. Toni has $5 to spend. The function y = 5 – 2x represents
the number of book covers x and notebooks y she can buy.
Graph the function. Interpret the points graphed.
Choose values for x and substitute
them in the function to find y.
Graph the ordered pairs (x, y).
She cannot buy negative amounts. So
she can buy 0 covers and 5 notebooks,
1 cover and 3 notebooks, or 2 covers
and 1 notebook.

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2. Graph y = x + 2.
Make a function table. Select any four
values for the domain x. Substitute these
values for x to find the value of y, and
write the corresponding ordered pairs.
Graph each ordered pair. Draw a
line that passes through each point.
The line is the complete graph
of the function. The ordered
pair corresponding to any point
on the line is a solution of the
equation y = x + 2.
Check It appears that (–2, 0) is also a
solution. Check this by substitution.
y = x + 2
0 = –2 + 2
0 = 0
Write the function.
Replace x with –2 and y with 0.
Simplify.
?

Course 3, Lesson
4-4
Functions
Words The value of y is one less than the corresponding value of x.
Equation y = x – 1
Ordered Pairs (0, –1), (1, 0), (2, 1), (3, 2)
Table Graph

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2
5. Each person that enters a store receives a coupon for $5 off
his or her entire purchase. Graph the function. Is the function
continuous or discrete? Explain.
Use the ordered pairs from the function table to
graph the function.
There can only be a whole number amount of
customers. The function is discrete. So, the
points are not connected.

To
• compare two functions represented in
different forms
Course 3, Lesson
4-5
Functions

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3
4
5
1. A zebra’s main predator is a lion. Lions can
run at a speed of 53 feet per second over
short distances. The graph at the right shows
the speed of a zebra. Compare their speeds.
To compare their speeds, compare the rates of change.
A zebra can travel at a rate of 59 feet per second. Since 59 > 53,
the speed of a zebra is greater than the speed of a lion.
A lion can travel at a rate of 53 feet per second.
To find the rate of change for a zebra, choose two points
on the line and find the rate of change between them.

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5 6
2. The function m = 140h, where m is the miles traveled in h hours,
represents the speed of the first Japanese high speed train. The
speed of a high speed train operating today in China is shown in the
table. Assume the relationship between the two quantities is linear.
Compare the y-intercepts.
Since 217 > 140, the function representing the Chinese high speed
train has a greater rate of change than the function representing the
Japanese high speed train.
Compare the rates of change.
Use the table to find the speed of the Chinese train.
a. Compare the functions’ y-intercepts and rates of change.
The speed of the Chinese train is or 217 miles per hour.
Find the distance on the
Japanese train.
m = 140h
m = 140(5)
m = 140h
Write the function.
Replace h with 5.
Simplify.
b. If you ride each train for 5 hours, how far will you travel on each?
7 You will travel 700 miles in 5 hours on the Japanese train.
Find the distance on the Chinese train by extending the table.
You will travel 1,085 miles in 5 hours on the Chinese train.
At 0 hours, no distance has been covered.
So, the y-intercepts are the same, 0.
The speed of the Japanese train is 140 miles per hour.

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5
6
3. Financial Literacy Angela and Benjamin each have a monthly
cell phone bill. Angela’s monthly cell phone bill is represented
by the function y = 0.15x + 49, where x represents the minutes
and y represents the cost. Benjamin’s monthly cost is shown
in the graph.
The rate of change for Angela’s monthly bill is $0.15 per minute.
Find the rate of change for Benjamin's bill.
a. Compare the y-intercepts and rates of change.
Angela’s monthly cost is represented by y = 0.15x + 49.
At 200 minutes, Angela will pay 0.15(200) + 49 or $79.
b. If you ride each train for 5 hours, how far will you travel on each?
The rate of change for Benjamin’s bill is $0.10 per minute.
So, Angela pays more per minute than Benjamin.
Use the graph to find Benjamin's cost. At 200 minutes,
Benjamin will pay $80.
The function for Angela’s bill has a y-intercept of 49. You can see from the graph that the
function for Benjamin’s bill has a y-intercept of 60. So, Benjamin has a greater initial cost.

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4. Financial Literacy Lorena’s mother
needs to rent a truck to move some
furniture. The cost to rent a truck
from two different companies is
shown in the table and graph.
Which company should she use
to rent the truck for 40 miles?
After 40 miles, the cost will be $75 + $25 or $100.
Find the cost of renting a truck from
Ron’s Rentals by extending the table.
The equation y = 0.5x + 30 where y represents the total cost and x represents
the miles driven can be used to find the total cost of renting the truck. After 40
miles, the cost will be 0.5(40) + 30 or $50. So, Cross Town Movers would cost
less for 40 miles.
10 25
20 50
30 75
The slope or rate of change is or 0.5.
Find the cost of renting a truck from Cross Town Movers by
analyzing the graph. The y-intercept of the graph is 30.

To find and interpret the
• rate of change of a function,
• initial value of a function
Course 3, Lesson
4-6
Functions

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4
1. A shoe store offers free points when you sign
up for their rewards card. Then, for each pair of
shoes purchased, you earn an additional
number of points. The graph shows the total
points earned for several pairs of shoes. Find
and interpret the rate of change and initial value.
To find the rate of change, choose two points from the graph.
Next find the initial value or the y-value when x = 0. Recall this value is called
the y-intercept. Extend the line so it intersects the y-axis. The value for y when
x = 0 is 30. So, the initial number of points earned is 30.
The rate of change is 15, so the amount of points earned per
pair of shoes is 15.

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5
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2. Joan has some photos in her photo album. Each week she plans to add
12 photos. Joan had 120 photos after 8 weeks. Assume the relationship
is linear. Find and interpret the rate of change and initial value.
Since each week Joan adds 12 photos to her photo album the rate
of change is 12. To find the initial value, use slope-intercept form
to find the y-intercept.
The y-intercept is 24. So, the initial amount of photos is 24.
y = mx + b Slope-intercept form
y = 12x + b Replace m with the rate of change, 12.
120 = 12(8) + b Replace y with 120 and x with 8
24 = b Solve for b.

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3. The table shows how much money Ava has
saved. Assume the relationship between the two
quantities is linear. Find and interpret the rate of
change and initial value.
Choose any two points from the table to find the rate of
change. The rate of change is or 20, so Ava
saves $20 each month.
To find the initial value, use the slope-intercept form to find
the y-intercept.
y = mx + b Slope-intercept form
y = 20x + b Replace m with the rate of change, 20.
110 = 20(3) + b Use the point (3, 110). x = 3, y = 110
50 = b Solve for b.
The y-intercept is 50, so Ava had initially saved $50.

To
• identify linear and non linear functions
Course 3, Lesson
4-7
Functions

1
2
1. Determine whether the table represents
a linear or nonlinear function. Explain.
As x increases by 2,
y decreases by 15 each time.
The rate of change is constant,
so this function is linear.

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3
2. Determine whether the table represents
a linear or nonlinear function. Explain.
As x increases by 3, y increases
by a greater amount each time.
The rate of change is not a constant,
so this function is nonlinear.
Graph the points on a coordinate plane.
Check
The points do not fall in a line. The function is nonlinear.

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4
3. Use the table to determine whether the minimum
number of Calories a tiger cub should eat is a
linear function of its age in weeks.
Use the table to find the rates of
change.
1,000 – 825 = 175
The rates of change are not the same. Therefore, this
function is nonlinear.
Check Graph the data to verify the ordered pairs do not lie on a
straight line.
1,185 – 1,000 = 185
1,320 – 1,185 = 135
1,420 – 1,320 = 100

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4. A square has a side length of s inches. The area of
the square is a function of the side length. Does this
situation represent a linear or nonlinear function?
Explain.
Make a table to show the area of the square for side
lengths of 1, 2, 3, 4, and 5 inches.
Graph the function. The function
is not linear because the points
(1, 1), (2, 4), (3, 9), (4, 16), and
(5, 25) are not on a straight line.

To
• graph a quadratic function,
• use the graph to estimate a value
Course 3, Lesson 4-
8
Functions

1
2
3
1. Graph y = x2.
To graph a quadratic function, make a table of values, plot the
ordered pairs, and connect the points with a smooth curve.

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2
2. Graph y = –x2 + 4.

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3
4
3. The function d = 4t2 represents the distance d in feet that a race car
will travel over t seconds with a constant acceleration of 8 feet per
second. Graph the function. Then use the graph to find how much
time it will take for the race car to travel 200 feet.
Time cannot be negative, so only use positive values of t.
Locate 200 on the vertical axis. Move over to the graph and
locate the corresponding value for the time.
The car will travel 200 feet after about 7 seconds.

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4. The function h = 0.66d 2 represents the distance d in miles you can
see from a height of h feet. Graph this function. Then use the graph
to estimate how far you could see from a hot air balloon 1,000 feet
in the air.
Distance cannot be negative, so use only positive values of d.
At a height of 1,000 feet, you could see approximately 39 miles.

To
• analyze qualitative graphs,
• sketch qualitative graphs
Course 3, Lesson
4-9
Functions

1
1. The graph below displays the water level in a bathtub. Describe the
change in the water level over time.
At time zero, the water level in the bathtub is zero. The water level
in the bathtub increases at a constant rate. Then the water is
turned off and the water level does not change. Finally, the drain
plug is pulled and the water level decreases at a constant rate until
the water level is zero.
increasing
decreasing
no change

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2. A tennis ball is dropped onto the floor. On each successive
bounce, it rebounds to a height less than its previous bounce
height until it comes to rest on the floor. Sketch a qualitative graph
to represent the situation.
Draw the axes. Label the vertical
axis “Distance from Floor.” Label
the horizontal axis “Time.”
Sketch the shape of the graph.
The distance from the floor
starts out at a high value. The
ball falls to the floor, bounces,
and rebounds to a height less
than its drop height. This pattern
is repeated several times until the
ball comes to rest on the floor.

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3. A child swings on a swing. Sketch a qualitative
graph to represent the situation.
Draw the axes. Label the vertical
axis “Distance from the ground”
and the horizontal axes “Time
Elapsed.”
Sketch the shape of the graph.
The distance from the swing to
the ground starts at a low value.
The child continues to swing and
creates momentum each time the
swing goes back until the child on
the swing stops.

(8) Chapter 4 Study Guide

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(8) Chapter 4 Study Guide