BLoCK 4 ~ ProPortIonALItY
direct variatiOn
L esson 20
L esson 21
L esson 22
L esson 23
L esson 24
L esson 25
r eview
THe coordinaTe PLane ----------------------------------------------Explore! Connect-The-Dots
M aking sense oF graPHs --------------------------------------------Explore! Train Trip
direcT variaTion TaBLes and graPHs --------------------------------direcT variaTion eqUaTions -----------------------------------------r ecognizing direcT variaTion --------------------------------------Explore! Graphs of Functions
wriTing L inear eqUaTions -------------------------------------------Explore! Jeremiah’s Journey
BLock 4 ~ direcT variaTion ------------------------------------------
word wAll
origin
QuAdr Ants
scAtte
e
sloP
inPu
t- o
Funct
u tP
ut
tABl
l ineA r
P l ot
n
Functio
y-Axis
ion
x
-A xis
120
e
r
Block 4 ~ Proportionality ~ Direct Variation
di r e
ct
VAri
Atio
n
122
127
133
140
146
153
160
BLoCK 4 ~ dIreCt VArIAtIon
tic - tac - tOe
Bean BaG ToSS
TraSh
r ectAngle dimensions
Design a carnival game
for bean bags using the
coordinate plane.
How much trash does
your family generate?
Look at Oregon rates.
Explore inverse variation by
finding areas for rectangles.
See page  for details.
See page  for details.
See page  for details.
PlAyground grAPhs
VAries directly ...
circumFerence
Think of 5 playground
activities. Graph a child’s
distance from the
ground over time
or speed over time.
Solve direct variation
problems using proportions
and y = mx.
Investigate how the
circumference of a circle
models direct variation.
See page  for details.
See page  for details.
See page  for details.
storewide sAle!
distAnce And sPeed
A like And unlike
Find discounts and sale
prices for items at a
department store and
how they are related
to direct variation.
Write a story that involves
both distance from home
and speed. Draw graphs
to match the story.
Explain the similarities
and differences of direct
variation functions
and linear functions.
See page  for details.
See page  for details.
See page  for details.
Block 4 ~ Direct Variation ~ Tic - Tac - Toe
121
the cOOrdinate Plane
Lesson 20
The coordinate plane is created by two number lines. The horizontal
y-axis
number line is called the x-axis and the vertical number line is called
the y-axis. The two number lines cross each other at zero. This point
of intersection is called the origin.
Quadrant
II
Quadrant
I
Origin
The x-axis and the y-axis divide the coordinate plane into four parts.
The parts are called quadrants. Starting at the top right quadrant and
moving counter-clockwise, they are named Quadrant I, Quadrant II,
Quadrant III and Quadrant IV.
Quadrant
III
x-axis
Quadrant
IV
Each point on a coordinate plane is identified by an ordered pair which has two numbers inside parentheses.
The first number represents the x-coordinate so it relates to the x-axis. The second number represents
the y-coordinate so it relates to the y-axis. The origin is the point (0, 0) because it occurs at 0 on both
the x-axis and the y-axis.
examPle 1
graph each point and name the quadrant where each point is located.
a. A(4, 6)
solutions
b. B(3, −2)
Quadrant Location
a. Quadrant I
b. Quadrant IV
c. None (on the y-axis)
d. Quadrant II
c. C(0, −8)
d. d(−5, 7)
A
d
B
C
examPle 2
solutions
122
Choose the letter with the given ordered pair.
a. (5, 0)
d. (8, −7)
b. (−4, 6)
e. (0, 3)
c. (−5, −2)
f. (0, 0)
a. U
d. T
b. V
e. S
c. R
f. P
Lesson 20 ~ The Coordinate Plane
V
s
r
P
u
t
exPlOre!
cOnnect-the-dOts
Kayla and Kelly like to play connect-the-dots. Kelly created her own pattern on a coordinate grid.
She gave the picture to Kayla to draw. She listed eight ordered pairs for Kayla to graph.
(5, 0) → (4, 3) → (3, 4) → (0, 5) → (−3, 4) → (−4, 3) → (−5, 0) → (0, −9)
step 1: Draw and label the x-axis and the y-axis. Label each one
from 0 to 10.
step 2: Draw the picture Kelly made by plotting the points. Connect
the dots in the order they were given. Connect the last point
to the first point.
step 3: Draw an additional line from (−5, 0) to (5, 0).
step 4: What picture did Kelly create?
step 5: Create your own picture of connect-the-dots on a coordinate grid using at least eight points.
Use points in all four quadrants. Two of the points must be on the x-axis or y-axis. Be sure to
list the points in the order they should be connected.
step 6: Switch lists of ordered pairs with a classmate. Try your classmate’s connect-the-dots puzzle.
Sometimes ordered pairs on a coordinate plane are listed in a table. The number in the x column is the
x-coordinate and the number in the y column is the y-coordinate.
examPle 3
solutions
use the table of values listed to the right.
a. Write the ordered pairs from the table using parentheses.
b. graph the ordered pairs on a coordinate plane.
c. Based on your graph, if the x-coordinate is 6, what will the
y-coordinate be?
a. (−3, −6), (−2, −4), (−1, −2), (0, 0), (1, 2), (2, 4), (3, 6)
b.
x
y
−3
−6
−2
−4
−1
−2
0
0
1
2
2
4
3
6
c. (6, 12) will be a point on the graph based on the pattern.
Lesson 20 ~ The Coordinate Plane
123
A scatter plot is a set of ordered pairs graphed on a coordinate plane. The data may be in a table or listed
as ordered pairs. It may represent real-life data. For example, it may compare age to height or time to rental
prices. It is often easier to determine if a relationship exists between data if the data is seen on a graph.
examPle 4
a. Make a scatter plot of the data. Label the x-axis “age” and the y-axis “height.”
Age (years), x
height (feet), y
2
5
7
11
13
18
2.5
4
4.5
5.2
5.7
6
b. Is there a relationship between the age of a person and that person’s height?
c. do you think this relationship will continue until someone is 70 years old?
explain.
a.
Height
solutions
Age
b. Yes, as the age increases so does the height. The points are going up from left to right.
c. No. People stop growing. In fact, a 70 year old may be shorter at age 70 than at age 18.
exercises
1. Draw a coordinate plane with an x-axis and a y-axis that go from −10 to 10. Label each axis, quadrant
and the origin.
2. On the coordinate plane drawn in Exercise 1, graph and label the following ordered pairs.
L(−6, −8)
A(5, −7)
U(4, 0)
G(2, 10)
3. Write the ordered pair for each point on the coordinate plane below.
B
C
A
d
e
F
124
Lesson 20 ~ The Coordinate Plane
g
H(−9, 1)
graph the ordered pairs. Connect the points in the order given. Connect the last point to the first.
name the figure.
4. (0, 0) → (4, 0) → (4, 4) → (0, 4)
5. (−3, −8) → (6, −8) → (0, 2)
6. (−3, −5) → (6, −5) → (6, 2) → (−3, 2)
7. (−1, 4) → (4, 4) → (7, −2) → (−4, −2)
ABCd is a rectangle. Find the ordered pair for point d.
8. A(0, 0), B(5, 0), C(5, 2), D(?, ?)
9. A(−3, −5), B(3, −5), C(3, 5), D(?, ?)
10. Write the ordered pairs for four points that are 4 units from the origin.
11. Choose the Quadrant I coordinate plane below that would best graph the ordered pairs. Explain why.
(0, 5), (2, 10), (4, 20), (5, 30), (6, 40), (8, 50)
A. 10
B. 60
10
10
C. 100
d. 50
50
100
Write the ordered pairs from each table. graph the points.
12.
x
y
−2
13.
x
y
−6
−2
−1
−3
0
1
2
14.
x
y
10
1
0
−1
5
2
4
0
0
0
3
2
3
1
5
4
5
6
2
10
5
6
Lesson 20 ~ The Coordinate Plane
125
15. Look at the following pattern.
Stage 0
Stage 1
Stage 2
Stage 3
Stage 4
a. For each numbered picture, count the number of squares and record it in the table.
stage number, x
0
1
2
3
4
number of squares, y
b. Graph the ordered pairs in the table to make a scatter plot.
c. Is there a relationship between the number of the figure and the number of squares
in the pattern? Explain.
d. Predict which figure number will have 19 squares.
16. Marla opened a bakery to sell homemade breads. The table below shows the average number
of loaves of bread she sold each day her first few months of operation.
number of months since the bakery opened, x
Average number of loaves sold each day, y
1
2
3
4
5
6
7
200
212
250
400
425
500
550
a. Draw a coordinate plane and label the x-axis and the y-axis.
Determine what the scale of each axis should be.
b. Graph the ordered pairs to make a scatter plot.
c. What can you conclude about Marla’s business based
on the points in the scatter plot?
review
solve each proportion.
x =_
5
17. __
12 6
x = __
4
18. __
20 16
6 =_
5
19. __
30 a
20. The Peterson family went out to dinner on Friday night. The bill was $42. They left a 15% tip for
their waitress.
a. How much did they leave for a tip?
b. What was the total cost of the meal and tip?
21. Owen bought a cowboy hat at the mall. The hat was
originally priced at $26. It was on sale for 30% off.
What did Owen pay for the cowboy hat?
22. Uma bought a cell phone in Washington. The cell phone
cost $99. She had to pay a 7.8% sales tax. How much did
she pay altogether for the cell phone?
126
Lesson 20 ~ The Coordinate Plane
making sense OF graPhs
Lesson 21
L
iz drove 150 miles to Mt. Hood at a constant speed. It took her three hours to get there. She stayed and
skied for six hours before driving home. Her drive home only took 2 _12 hours at a constant speed.
A graph can be drawn that models Liz’s trip. Let the x-axis represent time (in hours). Let the y-axis represent
Liz’s distance from home in miles. The important ordered pairs for this situation are described below.
description
ordered Pair
(hours, miles)
Liz started her trip 0 miles from home. No time
had passed.
(0, 0)
Three hours after leaving home, Liz was 150 miles
from home.
(3, 150)
Liz remained 150 miles from home until she left
Mt. Hood 9 hours after leaving home.
(9, 150)
Liz returned home 11.5 hours after starting
her trip.
(11.5, 0)
(9, 150)
Distance from home
(3, 150)
(11.5,0)
(0,0)
Number of hours
examPle 1
a. Find the rate of Liz’s trip to Mt. hood in miles per hour.
b. Find the rate of Liz’s trip home from Mt. hood in miles per hour.
solutions
150 miles = ______
50 miles or 50 miles per hour.
a. Liz drove 150 miles in 3 hours → _______
3 hours
1 hour
150 miles = ______
60 miles or 60 miles per hour.
b. Liz drove 150 miles in 2.5 hours → _______
2.5 hours
1 hour
Lesson 21 ~ Making Sense Of Graphs
127
Tamira rode the Ferris wheel at the state fair. The Ferris wheel made three revolutions
before stopping. Which graph below best represents Tamira’s height above the ground
based on the number of minutes she was on the ride?
B.
Height
Minutes
solutions
C.
Height
A.
Height
examPle 2
Minutes
Minutes
Graph C best shows Tanya’s height after each minute.
She goes up, comes down, then goes up again around the wheel.
Graph A looks like a Ferris wheel, but the graph goes up at first
and then goes backward to go down (goes left to go down).
This does not make sense because time does not go backward
on a Ferris wheel ride.
Graph B shows her getting to the top and staying at the same
height for a long time before coming down at a different rate.
It also shows her stopping before three times around.
exPlOre!
train triP
Mike took a train trip to visit his grandparents. The train made one stop before his stop in his
grandparent’s town.
◆ The train traveled at a speed of 50 miles per hour for 2 hours before stopping at the first stop.
◆ The train stopped for one hour.
◆ The train traveled at a speed of 40 miles per hour for 30 minutes before stopping at Mike’s stop
in his grandparent’s town.
step 1: How many miles did the train travel to the first stop?
step 2: How many miles did the train travel from the first stop to the last stop?
step 3: How long was Mike’s trip on the train from the time he started to the time he got off
the train at the last stop? Write your answer using hours.
step 4: Draw a first quadrant graph.
◆ Label the x-axis with “Hours”. Label the y-axis with “Miles from Home”.
◆ What scale should you use for the x-axis?
◆ What scale should you use for the y-axis?
128
Lesson 21 ~ Making Sense Of Graphs
exPlOre!
cOntinued
step 5: Model Mike’s train trip by graphing the points that represent Mike’s miles from home over time.
Label each point with the ordered pair (hours, miles from home).
Miles from Home
step 6: Mike’s grandmother picked him up at the train station. She had one errand to do on the way
to her house. The graph below represents his grandmother’s drive home from the train station.
Use the graph to answer the questions.
Hours
a. How far is the train station from his grandparent’s house?
b. When do they stop so grandma can run an errand? How long are they stopped for?
c. When are they driving the fastest? How do you know?
d. Find grandma’s rate of speed from 0.75 hours to 1 hour.
e. What does the point (1, 0) represent in this situation?
exercises
1. Tyler went jogging after school. He started at a quick pace and then slowed down over time,
Time
Distance from home
Distance from home
Distance from home
but continued to always go away from his home.
a. Which graph best represents Tyler’s distance from home over time? Why?
A.
B.
C.
Time
Time
C.
Speed
Speed
Speed
b. Which graph best represents Tyler’s speed over time? Why?
A.
B.
Time
Time
Time
Lesson 21 ~ Making Sense Of Graphs
129
2. Caroline climbs a hill next to her house at a steady pace and then runs down.
C.
Distance from home
B.
Distance from home
A.
Time
Distance from home
a. Which graph best represents Caroline’s distance from home over time? Why?
Time
Time
b. Which graph best represents Caroline’s speed over time? Why?
C.
Speed
Speed
B.
Speed
A.
Time
Time
Time
3. Latrelle climbed to the top of the play structure, sat at the top for awhile and then jumped down.
Time
C.
Distance from ground
B.
Distance from ground
A.
Time
Distance from ground
Which graph best represents Latrelle’s distance from the ground over time? Why?
Time
Choose the best story for each graph. explain how that choice fits the story.
4.
Speed
A. A bus pulls over at a bus stop.
B. A runner sprints to finish a race.
C. A person walks at a constant rate.
5.
Distance from home
Time
Time
6.
Speed
A. A car is stopped at a stoplight and then begins to move.
B. A car travels at a constant rate and then slows to a stop.
C. A plane sits on a runway before take-off.
Time
130
A. A girl goes to her friend’s house, stays for a
while and then returns home.
B. A car speeds up, travels at a constant speed
and then slows down.
C. A boy climbs up a ladder and immediately climbs down.
Lesson 21 ~ Making Sense Of Graphs
draw a graph for each story. Label the x-axis and y-axis.
7. Micah walks to the video store, browses through the movies at the store and then walks home. Graph
time on the x-axis and distance from home on the y-axis.
8. Sandra drives to the mall. Halfway there she realizes she forgot her purse so she returns home, grabs
her purse, and drives to the mall. Graph time on the x-axis and distance from home on the y-axis.
9. A person riding a bike is riding at a constant speed and then slows down to stop at a stop sign. Graph
time on the x-axis and speed on the y-axis.
10. Lyle drove to his cousin’s house, stayed for dinner and then drove home. The graph below shows his
(1, 60)
(4, 60)
Distance from home (miles)
distance from home (miles) over time (hours).
a. How far away does Lyle’s cousin live?
b. How long did Lyle stay at his
cousin’s house?
c. Find Lyle’s rate of speed in miles per
hour for the trip to his cousin’s house.
d. Find Lyle’s rate of speed in miles per
hour for the trip home from his
cousin’s house.
(5.5, 0)
Time (hours)
11. Kylie went running before school. The graph at right shows her
(1, 8)
(8, 8)
(11, 6) (14, 6)
Speed (mi/hr)
speed in miles per hour over time in minutes.
a. What was Kylie’s fastest speed?
b. How long did Kylie run at her fastest speed?
c. How many minutes did Kylie stop and rest before
running again?
d. What was Kylie’s fastest speed after stopping to rest?
e. How long did Kylie run at the speed in part d?
(8.5, 0)
(10, 0)
(14.5, 0)
Time (minutes)
12. Tatyana walked to the store at a rate of 4 miles per hour. She walked for one hour to get to
Distance from
home (miles)
the store. She shopped at the store for one hour before returning home walking at a rate
of 4 miles per hour.
a. How many miles is the store from Tatyana’s home?
b. Graph Tatyana’s trip using the coordinate plane at the right.
Be sure to include all ordered pairs when line segments
change direction.
Time (hours)
Lesson 21 ~ Making Sense Of Graphs
131
13. Vanderbilt rode his bike to his friend’s house. He traveled at a rate of 10 miles per hour for 30 minutes
Speed (miles per hour)
before taking 1 minute to slow down in order to stop and rest. He stopped for 3 minutes to rest. He then
took 1 minute to reach a speed of 6 miles per hour for the next 10 minutes. He took 1 more minute to
slow down and stop at his friend’s house.
a. How long after starting his trip did Vanderbilt stop to rest?
b. How long after starting his trip did Vanderbilt reach a speed of 6 miles per hour?
c. How long after starting his trip did Vanderbilt stop at his friend’s house?
d. Graph Vanderbilt’s trip using the coordinate plane below. Be sure to include all
ordered pairs when lines change direction.
Time (minutes)
review
14. Finn used 8 gallons of gas to travel 152 miles. How many gallons of gas will he use to drive 304 miles?
15. Aroldo ran 6 miles in 1.5 hours. At this rate, how long will it take him to run 10 miles?
draw a coordinate plane with an x–axis from –10 to 10 and a y-axis from –10 to 10. Plot each ordered pair
on the coordinate plane.
16. A(6, 0)
17. B(3, 5)
18. C(0, −3)
19. D(−7, 9)
20. E(−10, −4)
21. F(0, 0)
22. G(6, −5)
23. H(−8, 0)
24. I(5, 2)
t ic -t Ac -t oe ~ P l Ayg rou n d g r A P h s
Think of five different playground activities (slide, jump rope, climbing,
swing,…). For each activity, draw a graph of a child’s distance from the
ground over time or their speed over time. Be sure to include at least two
of each type of graph.
Explain each piece of your graph related to the playground activity.
Display the graph and its explanation on a poster to share with the class.
132
Lesson 21 ~ Making Sense Of Graphs
direct variatiOn taBles and graPhs
Lesson 22
W
hen a graph on a coordinate plane is a straight line that goes through the origin it is called a
direct variation graph. In this lesson you will investigate direct variation tables and graphs in Quadrant I.
As the x-coordinate gets larger, the y-coordinate also gets larger at a constant rate.
For example, Stein walks 4 miles per hour in a race from
Portland to the coast.
The table shows some ordered pairs describing his time since
the race started and the number of miles he has walked.
This is a direct variation graph because it goes through the point (0, 0)
and is a straight line.
Cathy earns $10 per hour babysitting. If Cathy works for 0 hours, she
earns $0. This means (0, 0) is a point on the coordinate plane that fits
the problem. Since (0, 0) is a point on the graph and the graph increases
at a constant rate of $10 per hour, a graph showing her earnings models
direct variation.
Money Earned
Miles, y
0
0
1
4
2
8
3
12
4
16
5
20
Miles
For each step the x-values increase by 1 and the y-values
increase by 4. The direct variation graph shows his
distance from the starting line in miles over time
in hours.
hours, x
Hours
hours Babysitting, x
Money earned, y
0
0
1
10
2
20
3
30
4
40
5
50
6
60
Hours
You can also determine if Cathy’s earnings show direct variation by looking at the table. Find the ratio of
y-coordinate
money earned to hours worked _________
for each ordered pair. If the ratios are equal, the table shows
x-coordinate
direct variation.
(
)
Lesson 22 ~ Direct Variation Tables And Graphs
133
hours Babysitting,
x
Money earned,
y
y-coordinate
__________
x-coordinate
0
0
Can’t divide by 0
1
10
10
__
1
2
20
3
30
4
40
5
50
6
60
20 __
__
= 10
2
1
30 __
10
__
=
3
1
40 __
10
__
= 1
4
50 __
__
= 10
5
1
60 __
10
__
= 1
6
For each ordered pair, the unit rate is $10 per hour, which matches Cathy’s babysitting rate.
The table shows direct variation.
For example, if Cathy works 1.5 hours, the point (1.5, 15) means that
after 1.5 hours of babysitting, Cathy earned $15. This point, however,
is not on the original scatter plot. If the points in the table are
connected with a line, the point (1.5, 15) will be a point on the line.
Money Earned
Sometimes it makes sense to connect the points on a scatter plot
with a line.
Hours
134
Lesson 22 ~ Direct Variation Tables And Graphs
determine whether or not each table models direct variation.
a. x y
b. x y
examPle 1
0
0
0
0
1
2
1
1
2
4
2
2
3
6
3
4
4
8
4
5
y-coordinate value
solutions
Look at the ratio _____________
for each ordered pair in the table. This tells the rate at
x-coordinate value
which the graph is changing.
0
a. (0, 0) → _
0
You cannot divide by 0. However, (0, 0) must be a
point for any direct variation relationship. Look at
the remaining points to see if there is a constant rate
of change.
Yes, this table models direct variation. The rate is _21 for
every ordered pair. The graph goes through the origin.
The rate _21 means for every 2 units up you move on
the graph, you move 1 unit right.
b. No, this table does not model direct variation.
y
The rate _x is not the same for every ordered pair.
x
y
y
_
x
0
0
--
1
2
2
_
1
2
4
4 _
_
=2
2 1
3
6
4
8
6 _
_
=2
3 1
8 _
_
=2
4 1
x
y
y
_
x
0
0
--
1
1
1
_
1
2
2
2 _
_
=1
2 1
3
4
4
_
3
4
5
5
_
4
y
In some cases, a table may not list the origin as one of the ordered pairs. If the ratio of _x is the same for all
pairs listed in the table, then the origin will be a point on that line even if it is not listed.
x
y
y
_
x
2
3
3
_
2
4
6
6 _
_
=3
4 2
6
9
9 _
_
=3
6 2
8
12
12 _
__
= 32
8
10
15
15 _
__
=3
10 2
Lesson 22 ~ Direct Variation Tables And Graphs
135
determine if each graph models direct variation. explain why or why not.
examPle 2
a.
solutions
b.
c.
a. Yes, this is a line through (0, 0). The y-value is proportional to the x-value.
b. No, although this is a line, it does not pass through the origin.
c. No, although the graph passes through the origin, it is not a line. The y-value is
not proportional to the x-value. In fact, the y-values increase at a greater rate
than the x-values increase.
exercises
each table below represents direct variation. graph each scatter plot. Find the rate by which each graph
increases.
1.
4.
136
x
y
x
y
x
y
0
0
1
3
0
0
1
2
0
0
2
1
2
6
2
4
4
2
3
9
3
6
6
3
4
12
4
8
8
4
x
y
x
y
x
y
3
1
6
2
1
4
0
0
2
8
2
1.5
9
3
3
12
4
3
12
4
4
16
6
4.5
15
5
5
20
8
6
2.
5.
Lesson 22 ~ Direct Variation Tables And Graphs
3.
6.
7. Kirsten walked 4 miles per hour for 5 hours to train for an upcoming marathon walk.
a. Copy and complete the table below.
number of hours Kirsten walked, x
0
1
2
3
4
5
number of miles Kirsten walked, y
b. Draw a scatter plot of the ordered pairs in the table. Does it make sense to connect the points
with a line? Explain why or why not.
c. Does this situation model direct variation? Explain why or why not.
8.
a. Draw the next two figures in the pattern below.
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
b. Complete the table showing the figure number and its corresponding number of squares.
Figure number, x
1
2
3
4
5
number of squares, y
c. Does this pattern model direct variation? Explain.
d. If the pattern models direct variation, find its rate.
determine whether the table models direct variation. explain why or why not. If it does, give the rate.
9.
12.
10.
11.
x
y
x
y
x
y
0
0
0
0
0
0
1
1
2
1
1
5
2
2
4
2
2
10
3
3
6
3
3
12
4
4
8
4
4
20
x
y
x
y
x
y
3
1
6
12
0
0
6
2
12
24
5
12
9
3
18
32
10
24
12
3
24
48
15
36
15
5
30
60
20
48
13.
14.
15. Draw a scatter plot that models direct variation. Be sure to include at least five points on your
coordinate plane. Identify the points by labeling them as ordered pairs on the graph.
16. Explain how you can tell whether or not a table shows direct variation.
Lesson 22 ~ Direct Variation Tables And Graphs
137
determine whether the graph models direct variation. explain why or why not. If it does, give the rate.
17.
18.
(4, 8)
(1, 6)
(3, 4)
(1, 2)
20.
19.
(2, 1)
(8, 5)
(4, 2)
(4, 3)
21. Explain how you can tell whether or not a graph models direct variation.
review
Write as a unit rate.
6 miles
22. ______
2 hours
24 calls
23. _______
0.5 hours
25. Find each probability for one roll of a number cube.
a. P(4)
b. P(odd number)
c. P(not 1)
26. You choose a letter at random from the word SUMMER. Find each
probability.
a. P(M)
b. P(consonant)
c. P(X)
27. The probability of picking a green jelly bean out of a bag is _13 . There are
72 jelly beans in the bag. Predict how many are green.
28. The probability that Mr. Jenkins picks a boy to answer a question in class
is 2 out of 5. If he picks 15 students to answer questions today, how many
would you predict will be boys?
138
Lesson 22 ~ Direct Variation Tables And Graphs
110 steps
24. _______
4 minutes
t ic -t Ac -t oe ~ r e c tA ngl e d i m e n s ion s
A rectangle has an area of 24 square feet.
1. Find all the possible lengths and widths for the rectangle that are
whole numbers. Copy and complete the chart with the information.
Length (ft), x
Width (ft), y
2. Graph the ordered pairs (length, width) in Quadrant I. Connect the points with a smooth curve.
3. The graph does not show direct variation. It shows inverse variation. As the x-values increase,
24
the y-values decrease. It has the equation y = __
x since x ∙ y = 24 . Explain how you can tell from
24
__
the equation that y = x is not direct variation.
4. Suppose you need all the rectangles whose areas equal 36 square inches. Write the inverse
variation equation which gives all ordered pairs (length, width) with that area. Create a graph
of this relationship. Connect the points with a smooth curve.
5. Explain why the graph can be represented by a smooth curve rather than just as points. Are there
any limits to this curve on the graph? Why might this be the case?
t ic -t Ac -t oe ~ d i s tA nc e
And
sPeed
A little boy traveled home after his first day of kindergarten. He traveled
at least three different speeds on his way home. He stopped at least once
during his trip.
Write a story about the little boy’s trip home. Explain how he traveled from school
to home. Be creative and make it as adventurous as possible.
Draw a graph that shows his distance from home over time. Draw a second graph
showing his speed over time. Label the axes and include ordered pairs for important
changes in distance or speed on the graph.
Lesson 22 ~ Direct Variation Tables And Graphs
139
direct variatiOn e QuatiOns
Lesson 23
Money Earned
For every hour
I baby sit,
I make $10.
Hours
hours Babysitting, x
Money earned, y
0
0
1
10
2
20
3
30
4
40
5
50
6
60
In the last lesson, the information about Cathy’s babysitting job was displayed using a graph, a table and
words. A function is another way of displaying information. A function is a pairing of numbers according to
a specific rule or equation. An input-output table shows how each input in a function is paired with exactly
one output value. The input is usually represented by the variable x and the output is usually the variable y.
A function, or equation, can be written to help Cathy figure out how much money, y, she makes after x hours.
Look at these proportions:
y dollars
10 dollars ______
_______
=
1 hour
3 hours
y = 10 ∙ 3 = 30
y dollars
10 dollars ______
_______
=
1 hour
6 hours
y = 10 ∙ 6 = 60
y dollars
10 dollars ______
_______
=
1 hour
x hours
y = 10 ∙ x = 10x
Generally, a direct variation function has the form y = mx, where m is the constant unit rate of change. In a
y
direct variation function, the m is also called the slope of the line in the graph. Think about the ratio _x as
the constant rate of change, m. This means:
y
_
x =m
140
y
Multiply both sides by x.
x ∙ _x = m ∙ x
Simplify.
y = mx
Lesson 23 ~ Direct Variation Equations
examPle 1
use the direct variation function y = 3x.
a. Complete the table for the given input values.
Input, x
Function
rule
y = 3x
output, y
0
1
2
3
4
b. draw a scatter plot of the ordered pairs in the table and connect the ordered
pairs with a straight line.
c. Find the slope of the function.
solutions
a.
Input, x
Function
rule
y = 3x
output, y
0
3(0)
0
1
3(1)
3
2
3(2)
6
3
3(3)
9
4
3(4)
12
b.
c. The slope is the coefficient of the x variable.
y = 3x Slope = 3
Lesson 23 ~ Direct Variation Equations
141
examPle 2
solution
This table shows ordered pairs which model direct variation.
Write an equation relating the x and y coordinates.
x
0
2
4
6
8
y
0
1
2
3
4
y-coordinate
_________
.
x-coordinate
Find the slope by calculating the ratio
Use any ordered pair.
(2, 1) → m = _12
Write the equation in the form y = mx.
y = _12 x or y = 0.5x
exercises
Copy and complete each input-output table and graph each function in Quadrant I.
1.
3.
Input
x
Function
rule
y = 4x
output
y
2.
0
0
1
1
2
2
3
3
4
4
Input
x
Function
rule
y = 6x
output
y
4.
0
output
y
Function
rule
y = __31 x
output
y
Function
rule
y = __23 x
output
y
6
3
9
4
12
Function
rule
y = 2x
0
2
4
6
8
output
y
6.
Input
x
0
2
4
6
8
7. Find the slope for each direct variation equation in exercises 1-6.
142
Function
rule
y=x
3
2
Input
x
Input
x
0
1
5.
Input
x
Lesson 23 ~ Direct Variation Equations
8. Juanita opened her own movie theater. She plans to show older movies at a reduced price compared to
the large theater in town that shows new releases. She plans to charge $3.00 per person and hopes to fill
her 50-seat theater once in the late afternoon and once in the evening.
a. Complete this table to show how much money Juanita will get for
selling the given numbers of tickets to a show.
number of tickets sold, x
0
10
15
20
40
50
Money collected, y
b. Can this situation be modeled by direct variation? Explain why or
why not.
c. Write an equation for the amount of money collected (y) based on the
number of tickets sold (x).
d. If Juanita sells out both shows in the afternoon and evening, how much
money will she collect?
9. Bryan and Megan tried to write an equation for the direct variation relationship given in the table below.
Each student used a different method.
x
0
4
8
12
16
20
y
0
1
2
3
4
5
Look at both solutions and determine who is correct. For the student that made a mistake, identify the
mistake and explain what the student should have done instead to use their method correctly.
Bryan’s Work
Megan’s Work
Use the point (4, 1) to
find the constant
rate of change.
4=4
_
1
Use the point (4, 1)
to set up a proportion
to find the equation.
y
1=_
_
4 x
The equation is:
y = 4x
Use cross products.
4y = x
Divide both sides by 4.
x
y=_
4
The equation is:
1x
y = __
4
10. a. Draw the next two figures in the pattern below.
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
b. Complete the table showing the figure number and its corresponding number of squares.
Figure number, x
1
2
3
4
5
6
7
number of squares, y
c. Does the table show direct variation? Explain why or why not.
d. Write a function which gives the number of squares in the pattern based on the figure number.
e. Use your function to determine how many squares would be in Figure 25.
Lesson 23 ~ Direct Variation Equations
143
Find the slope of each direct variation equation.
11. y = 7x
12. y = 0.75x
13. y = 20x
graph each direct variation equation.
14. y = 4x
16. y = _34 x
15. y = x
17. Explain what the “m” represents in the direct variation equation y = mx.
This table shows ordered pairs which model direct variation. Write an equation relating the
x and y coordinates.
18.
x
y
0
0
1
19.
x
y
0
0
7
1
2
14
3
21
20.
x
y
0
0
2.5
2
16
2
5
5
40
3
7.5
6
48
review
21. A rabbit hops away from home for 1 hour at a rate of 30 hops every 10 minutes.
22. Choose the best description for the graph given.
A. As the distance from the sun increases, the intensity of light increases.
B. As the distance from the sun increases, the intensity of light decreases.
C. As the distance from the sun decreases, the intensity of light decreases.
23. Nigel jogs at a speed of 9 feet per second. How many feet does he jog in one hour?
Light Intensity
He then rests for _12 hour before returning home in 30 minutes at a rate of 30 hops
every 5 minutes.
a. How many hops did the rabbit make before stopping to rest?
b. How many hops did the rabbit make after resting to get home?
c. Did the rabbit hop at a faster rate leaving home or returning to home? Explain.
d. Draw a graph that shows the rabbit’s distance from home (number of hops)
based on the time since he started his trip. Be sure to label all axes and give
ordered pairs when lines change direction.
Distance from Sun
24. A bicyclist rides at a speed of 20 miles per hour.
a. How many feet does he ride in one hour?
b. How many feet does he ride in one minute?
25. Lucy takes 33 breaths per minute. How many breaths does
she take in one week?
144
Lesson 23 ~ Direct Variation Equations
t ic -t Ac -t oe ~ s t or e w i de s A l e !
A department store decided to reduce all prices
by 20% from 7:00 until 11:00 on Saturday to boost sales.
Samantha went shopping and bought the
following items.
1. Find the discount on each item. Add a column to the chart at right to
display your values.
2. Graph the ordered pairs (original price, discount). Connect the points to
show the direct variation.
3. Find the slope of the direct variation equation and write the equation.
4. Find the amount Samantha paid for each item. Add a fourth column to
the chart above to display the values.
5. Graph the ordered pairs (original price, new price) on a new coordinate plane.
Connect the points to show the direct variation.
6. Find the slope of the direct variation equation and write the equation.
7. Explain the relationship between the slopes in #3 and #6.
Item
original
Price
Shirt
$30
Pants
$50
Shoes
$60
Tie
$24
Jacket
$80
Purse
$36
t ic -t Ac -t oe ~ c i rc u m F e r e nc e
The circumference of a circle is the distance around its outside edge.
The equation to find the circumference of a circle is related to its diameter.
The larger the diameter, the larger the circumference.
diameter
Circumference = π ∙ diameter
C = πd
The symbol π is a decimal that extends forever and never repeats. Its approximate value is 3.14.
1. Find the circumference of each circle with the given diameter. Use 3.14 for π.
a. d = 2 feet
b. d = 5 inches
c. d = 10 meters
d. d = 20 centimeters
2. Find the circumference of each circle with the given radius. A radius is the distance from
the center of the circle to its outside edge.
a. r = 1 inch
b. r = 3.5 feet
c. r = 7 centimeters
d. r = 4 meters
3. Copy and complete the table with the information from #1 and #2. Explain how the table shows
direct variation. Find the ratio
diameter, x
2
5
y-coordinate
.
( _________
x-coordinate )
10
20
2
7
14
8
Circumference, y
4. Graph the ordered pairs in the table in #3 and connect them with a line. Explain how the graph
shows direct variation. Does the point (0, 0) fit on the graph? Explain why this makes sense.
5. Explain how the equation C = πd is a direct variation equation. Identify the slope.
Lesson 23 ~ Direct Variation Equations
145
recOgniZing d irect variatiOn
Lesson 24
D
irect variation functions can be graphed on a full coordinate plane using all four quadrants. The slope can
be positive or negative and the line must go through the origin.
exPlOre!
graPhs OF FunctiOns
There are many types of functions. In this activity you will graph a few of them and then look at input-output
tables to help figure out when a function is a direct variation function.
step 1: Copy and complete each input-output table.
A. Linear Function
B. direct Variation Function
Input
x
Function rule
y = 2x + 1
output
y
Input
x
Function rule
y = 3x
output
y
−3
2(−3) + 1
−5
−3
3(−3)
−9
−2
−2
−1
−1
0
0
1
1
2
2
3
3
C. Quadratic Function
d. Inverse Variation Function
Input
x
Function rule
y = x²
output
y
Input
x
−3
(−3)² = (−3)(−3)
9
−4
−2
−1
0
1
4
__
−4
output
y
−1
−2
−1
0
2
3
Function rule
y = __4x
Impossible.
Cannot divide
by 0.
1
2
4
step 2: Draw four coordinate planes. Each coordinate plane should have all four quadrants. Plot the
ordered pairs from each table in step 1 on a separate coordinate plane. Connect the points with
a smooth curve or line for graphs A, B and C. For graph D, connect the first three points with
one curve. Connect the last three points with a separate curve.
146
Lesson 24 ~ Recognizing Direct Variation
exPlOre!
cOntinued
step 3: Add a column to the end of each input-output table in step 1. Fill in the column with the ratio of
y-coordinate
the y-coordinate to the x-coordinate _________
for each ordered pair in the table.
x-coordinate
,
ple
m
a
x
A
re h
Fo rap
g
Linear Function
Input
x
Function rule
y = 2x + 1
output
y
y
_
x
−3
2(−3) + 1
−5
−2
2(−2) + 1
−3
−5 _
__
=5
−3 3
−3 _
__
=3
−2 2
y-coordinate
step 4: For which function(s) is the ratio _________ always equal for every ordered pair
x-coordinate
(except when x = 0)?
step 5: How can you tell if a function is direct variation when you look at the:
a. graph?
b. table of values?
c. equation?
Lesson 24 ~ Recognizing Direct Variation
147
examPle 1
tell whether or not each graph is a direct variation graph. explain your choice.
a.
b.
c.
d.
solutions
a. NOT direct variation. Although it is a line, it does not go through (0, 0).
b. NOT direct variation. Although it goes through (0, 0), it is not a line.
c. Direct variation. It is a line that goes through (0, 0).
d. Direct variation. It is a line that goes through (0, 0).
examPle 2
tell whether or not each equation is a direct variation equation. explain your choice.
If it is direct variation, identify the slope.
5
a. y = −2x
b. y = _
c. y = 4x − 3
x
d. y = _23 x
e. y = 3x²
f. y = 2x
solutions
a. Yes, direct variation: m = −2 so the slope is −2.
b. NOT direct variation because the x is in the denominator.
c. NOT direct variation because it has a “−3” at the end which means it
does not go through (0, 0).
d. Yes, direct variation: m = _23 so the slope is _23 .
e. NOT direct variation because the x has an exponent of 2.
f. NOT direct variation because the x is an exponent.
examPle 3
tell whether or not each table shows ordered pairs that model direct variation.
explain your choice. If the ordered pairs do show direct variation, identify
the slope.
a.
148
x
y
−2
b.
x
y
−10
−2
−1
−5
0
c.
x
y
−2
−2
−1
−1
0
−1
−2
0
0
2
1
2
1
5
1
4
2
1
2
10
2
6
4
1
_
2
Lesson 24 ~ Recognizing Direct Variation
examPle 3
solutions
a.
b.
c.
x
y
y
_
x
−2
−10
−10 _
___
= 51 = 5
−2
−1
−5
−5 _
__
=5=5
−1 1
0
0
--
1
5
5
_
=5
1
2
10
10 _
__
= 51 = 5
2
x
y
y
_
x
−2
−2
−2 _
__
=1=1
−2 1
−1
0
0
__
=0
−1
0
2
--
1
4
4
_
=4
1
2
6
6 _
_
=3=3
2 1
x
y
y
_
x
−2
−1
−1 _
__
=1
−2 2
−1
−2
−2 _
__
=2=2
−1 1
1
2
2
_
=2
1
2
1
1
_
2
4
1
_
2
1
_2 = _
4 8
Yes, the table shows direct variation.
y
The ratio _x = 5 for all ordered pairs
(except when x = 0) and the function goes
through (0, 0).
y
NOT direct variation. The ratio _x is not equal
for each ordered pair and the graph does not
go through (0, 0).
NOT direct variation. The ratio
for each ordered pair.
y
_
x
is not equal
1
__
exercises
tell whether or not each graph is a direct variation graph. explain your choice.
1.
2.
3.
4.
5.
6.
Lesson 24 ~ Recognizing Direct Variation
149
tell whether or not each equation is a direct variation equation. explain your choice. If it is direct variation,
identify the slope.
7. y = 2x
9. y = _2x
8. y = −0.2x
10. y = _6x
11. y = 3x²
12. y = _45 x
13. y = − _17 x
14. y = 5x
15. y = 1.5x
tell whether or not each table shows ordered pairs that model direct variation. explain your choice. If the
ordered pairs show direct variation, identify the slope.
16.
19.
x
y
−2
−12
−1
−6
0
0
1
6
2
12
x
y
−6
−3
−2
−9
−1
−18
1
18
2
9
17.
20.
18.
x
y
x
y
−2
−3
−1
1
−1
−6
0
0
1
6
1
−1
2
3
2
−2
3
2
3
−3
x
y
x
y
−2
−20
−3
9
−1
−10
−2
6
0
0
−1
3
1
10
1
−3
2
20
2
−6
21.
graph each direct variation equation on a coordinate plane with four quadrants.
23. y = _12 x
22. y = 3x
24. y = −x
25. Graph a direct variation function that goes through the point (−2, −6). What is the slope?
26. Juan took a test and finished three questions every 5 minutes.
a. Copy and complete the table below. The x-value is the time Juan spent on the test and the y-value is
the total number of questions he finished since he started.
Minutes, x
0
5
Questions finished, y
0
3
10
15
20
25
30
b. Does the table in part a show direct variation? Explain why or why not.
c. If the table shows direct variation, find the slope of the direct variation function.
d. If the table shows direct variation, write the direct variation equation y = mx and replace the m with
the slope from part c.
150
Lesson 24 ~ Recognizing Direct Variation
27. Marna rented a movie from the store. It costs $1.00 to rent and then costs an additional $1.00 for each
day Marna keeps the movie before returning it.
a. Copy and complete the table below. The x-value is the number of days Marna keeps the movie
and the y-value is the amount of money the movie costs her to rent.
days, x
0
1
Cost ($), y
1
2
2
3
4
5
6
b. Does the table in part a show direct variation? Explain why or why not.
c. If the table shows direct variation, find the slope of the direct variation function.
d. If the table shows direct variation, write the direct variation equation y = mx and replace the m with
the slope from part c.
A. Mark ran a race at a speed of 6 miles per hour.
B. Mark ran a race at a speed of 1 mile per hour.
C. Mark ran a race at one speed and then stopped for awhile before running again.
29. Choose the best explanation for the graph.
A. As the number of buses increases, the number of students on each bus increases.
B. As the number of buses increases, the number of students on each bus decreases.
C. As the number of buses decreases, the number of students on each bus decreases.
(2, 12)
Hours
Students on
each bus
28. Choose the best explanation for the graph.
Distance
review
(1, 60)
(2, 30)
Number of buses
determine the scale factor for each pair of similar figures.
30.
31.
4
10
32.
2
0.5
2
6
Lesson 24 ~ Recognizing Direct Variation
151
t ic -t Ac -t oe ~ VA r i e s d i r e c t ly ...
Direct variation functions come directly from proportions.
Sometimes proportion word problems include the words “varies directly”.
Example:
The amount of gas Kami’s car uses varies directly with the number of miles traveled.
If Kami used 3 gallons to travel 75 miles, how many gallons of gas will she use to
travel 125 miles? Two solutions are shown below.
use y = mx
use Proportions
Write a proportion.
3 gallons _______
y gallons
_______
=
75 miles 125 miles
*Use cross products.
Divide both sides by 75.
75y = 375
y=5
Let x = miles and y = gallons.
(75, 3) and (125, y) are two points on the graph.
Find m using known point (75, 3).
Substitute x = 3 and y = 75.
Divide both sides by 75.
3 = m ∙ 75
3
__
=m
75
Write the direct variation equation and solve.
3
Substitute m = __
.
75
3
y = __
x
75
Simplify.
y=5
*Substitute x = 125.
Kami will use 5 gallons of gas.
3
y = __
∙ 125
75
Kami will use 5 gallons of gas.
1. Explain how the two starred steps in the above table are equal.
use proportions or the equation y = mx to solve each problem. use each method once.
2. The weight of an object on the moon varies directly with its weight on Earth. Suppose an astronaut
weighs 240 pounds on Earth and 40 pounds on the moon. How much would a person who weighs
120 pounds on Earth weigh on the moon?
3. Javier’s wages vary directly with the number of hours he has worked. His wages for 6 hours are $48.
How much will he earn for 25 hours of work?
4. Write and solve two of your own direct variation problem. Each must contain the phrase
“varies directly.” Show how to solve each problem using proportions and the equation y = mx.
152
Lesson 24 ~ Recognizing Direct Variation
writing linear eQuatiOns
Lesson 25
In this block you have learned how to recognize direct variation graphs, tables and equations. Many
functions, when graphed, are straight lines. These functions are called linear functions. Direct variation
graphs pass through the origin, but most graphs of linear functions do not pass through the origin. In this
lesson you will learn how to develop an equation for any linear function from a graph or table.
exPlOre!
Jeremiah’s JOurney
Jeremiah went for a walk. Jeremiah graphed his distance from home in city blocks at
several points during his walk.
step 1: Copy and complete the input-output
table below with the five ordered pairs
shown on the graph.
Input
Minutes, x
output
Blocks from home, y
step 2: When the input column goes up by one each time, it can be called the counter column. Is this table’s
input column a counter column?
step 3: The equation will have a start value. The start value can be found by locating the output amount that
is paired with the input amount of zero. What is the start value for Jeremiah’s Journey chart? Fill
this start value into the developing equation.
y = _____
step 4: Look at the output column. Is this column increasing or decreasing as the counter column gets
bigger? How much is it increasing or decreasing for each step?
step 5: The amount of increase or decrease occurring in each step is called the rate of change. The rate is
always the coefficient of x in the equation. Fill in the amount from step 4 into the equation.
(NOTE: Choose the addition symbol if the output column increases and the subtraction symbol
if the output column decreases.)
y = _____ ± _____x
Lesson 25 ~ Writing Linear Equations
153
exPlOre!
cOntinued
step 6: In direct variation, you do not need to write the start value in the equation because the
start value is 0. Is this one of those situations? How do you know?
step 7: Use your equation to predict how many blocks from home Jeremiah was when his mother
picked him up 12 minutes later.
examPle 1
solution
Find the linear equation for the input-output table.
Input
x
output
y
−1
14
0
11
1
8
2
5
3
2
The start value is the output value
paired with zero. In this function,
the start value is 11.
Start the equation.
y = 11 ± ___x
The rate of change is determined
by the amount the output column
increases or decreases in one unit.
In this function, the rate of change
is −3. When the x-value increases by
one, the y-value decreases by 3.
Finish the equation. y = 11 − 3x
154
Lesson 25 ~ Writing Linear Equations
Input
x
output
y
−1
14
0
11
1
8
2
5
3
2
−3
−3
−3
−3
examPle 2
Find the linear equation for the graph.
solution
Create an input-output table using the x- and y-values from each point.
Input
x
output
y
−1
−6
0
−4
1
−2
2
0
3
2
4
4
5
6
+2
+2
+2
+2
+2
+2
The start value is the output value paired with zero.
In this equation, the start value is −4.
Start the equation.
y = −4 ± ____x
The rate of change is determined by the amount the output column increases or
decreases when the x-value increases by 1 unit. In this function, the rate of change
is +2.
Finish the equation.
y = −4 + 2x
Find the start value.
Then find the rate of
change.
Lesson 25 ~ Writing Linear Equations
155
exercises
Find the linear equation for the input-output tables.
1.
3.
Input
x
output
y
Input
x
output
y
0
17
−1
−4
1
12
0
−1
2
7
1
2
3
2
2
5
4
−3
3
8
5
−8
4
11
Input
x
output
y
−3
4.5
Input
x
output
y
−2
5.0
−2
−4
−1
5.5
−1
−2
0
6.0
0
0
1
6.5
1
2
2
7.0
2
4
3
6
2.
4.
5. Which table in exercises 1-4 models direct variation? How do you know?
6. Use the equation y = 3 + 4x.
a. What is the rate of change? How do you know?
b. What is the start value?
c. Is the graph of this function increasing or decreasing? How do you know?
7. Use the equation y = 3x.
a. What is the rate of change? How do you know?
b. What is the start value?
c. Is the graph of this function increasing or decreasing? How do you know?
d. What special type of linear function is y = 3x?
Find the linear equation for each graph.
8.
156
9.
Lesson 25 ~ Writing Linear Equations
10.
11.
12. Which graph in exercises 8-11 models direct variation? How do you know?
13. Karin makes bracelets during the summer. At the beginning of the summer she had made 2 bracelets.
Each day during the summer she made 3 more bracelets.
a. Copy and complete the input-output table based on the information above.
Input
days, x
output
Bracelets
Made, y
0
2
1
2
3
4
5
b. Find the equation for the table.
c. Use the equation to determine the total number of bracelets Karin will have made after 20 days
by substituting 20 for x.
14. Explain the first step to finding an equation when given a graph of the linear function.
15. Janelle came up with linear equations for two different problems. Her teacher told her the equations
were correct but could both be written in a simpler form. Write her two answers in simplest form.
Linear equation #1: y = 0 + 4x
Linear equation #2: y = 3 + 1x
16. Ollie started the school year with $430 which he saved from his summer job. Each week,
he spends $32 of his savings on snacks for himself and music for his MP3 player.
a. Copy and complete the input-output table based on the
Input
output
information above.
Weeks, x
Money
b. Find the linear equation for the table.
remaining, y
c.
Use the linear equation to determine how much money
0
Ollie will have left after 8 weeks by substituting 8 for x.
1
2
3
4
5
Lesson 25 ~ Writing Linear Equations
157
17. Find the missing values in each table. Write the linear equation for each table.
a.
Input
x
output
y
Input
x
output
y
−2
10
0
40
−1
12
1
36
2
32
b.
0
16
5
2
10
18. Saran did chores for her grandma. Her grandma deposited her pay in Saran’s savings account each week.
Saran checked the balance of her account 5 different times as seen in the table below.
a. What is the start value for the function representing Saran’s total savings?
What is the real-life meaning of the start value in this situation?
Weeks
Passed, x
b. Find the rate of change (the amount Saran is paid each week).
0
c. Explain in words how you determined the rate of change for this function.
d. Write the equation for the table.
3
e. Predict how much money will be in Saran’s account after 12 weeks.
4
Account
Balance, y
$40
$76
$88
7
$124
8
$136
review
tell whether each equation, graph or table represents a direct variation function. explain your answer.
19. y = 4x²
23.
22.
158
20. y = _14 x
Lesson 25 ~ Writing Linear Equations
21.
24.
x
y
−2
8
−1
4
0
0
1
−4
2
−8
x
y
0
5
1
10
2
15
3
20
4
25
t ic -t Ac -t oe ~ t r A s h
In 2006, Oregonians generated 3,118 pounds of waste per person
(BlueOregon.com/). Of that, 47.5% was recycled per person.
1. How many pounds of waste did each person recycle on average in 2006?
Round to the nearest pound.
2. How many pounds of trash (not recycled material) did each person
dispose of in 2006? Round to the nearest pound.
3. How much overall waste would a family of four have generated in 2006?
4. Write a direct variation equation for the total amount of overall waste, y, produced by x people.
5. Write a direct variation equation for the total amount of recycled waste, y, produced by x people.
6. Write a direct variation equation for the total amount of trash (not recycled), y, produced
by x people.
7. Which direct variation equation has the largest slope?
8. Use your equations to find the total amount of overall waste, recycled waste and trash each number
of people would have generated in 2006. Organize your information in a table.
a. 4 people
b. 100 people
c. 1,000 people
d. 1,000,000 people
e. Find the current population for Oregon. Use this number.
t ic -t Ac -t oe ~ A l i k e
And
un l i k e
Fold a piece of paper in half vertically. On one side label the column
‘Similarities’. Label the other column ‘Differences’.
Write the similarities and differences between direct variation functions and linear
functions. List as many in each column as you can. Be sure to include information
related to equations, tables and graphs. Use examples.
Lesson 25 ~ Writing Linear Equations
159
review
BLoCK 4
vocabulary
direct variation
function
input-output table
linear function
origin
quadrants
scatter plot
slope
x-axis
y-axis
Lesson 20 ~ The Coordinate Plane
Write the ordered pair for each point on the coordinate plane below.
1. M
A
2. A
M
t
3. T
4. C
5. H
C
h
graph the ordered pairs. Connect the points in the order given. Connect the last point to the first.
name the figure.
6. (0, 0), (5, 4), (−3, 2)
160
Block 4 ~ Review
7. (2, −3), (2, 4), (−4, 4), (−4, −3)
Write the ordered pairs from each table. Then graph the points.
8.
x
y
x
y
−3
−4
−3
5
−2
−2
−2
3
−1
−3
−1
1
0
0
0
−1
1
2
1
−3
2
3
2
−5
3
5
3
−7
9.
10. Marsella counted the number of cars at an intersection every minute. She recorded the minutes
since 7 o’clock and the total number of cars since 7 o’clock in a table.
Minutes since 7:00, x
0
1
2
3
4
total number of cars, y
0
4
9
13
16
a. Plot the ordered pairs to make a scatter plot in Quadrant I.
b. Looking at the graph, estimate how many cars will go through the intersection by 7:10.
Lesson 21 ~ Making Sense of Graphs
B.
C.
Distance from ground
A.
Distance from ground
11. A person jumps on a trampoline.
Distance from ground
Choose the best graph for each story. explain how your choice fits the story.
Time
Time
Time
12. Othello climbs up a ladder at a constant speed, stops to rest and then keeps climbing up.
Time
Distance from ground
Distance from ground
Distance from ground
a. Which graph best represents his distance from the ground over time? Why?
A.
B.
C.
Time
Time
Block 4 ~ Review
161
Time
Speed
C.
Speed
Speed
b. Which graph best represents his speed over time? Why?
A.
B.
Time
Time
Choose the best story for each graph. explain how your choice fits the story.
13.
Amount
A. Amount of candy left in a dish set out at a party.
B. Amount of rain each hour during a storm.
C. Total amount of rain during a storm.
Time
A. A person drives toward home.
B. A person walks and then runs away from home.
C. A person runs and then slows down but continues
traveling awayfrom home.
Distance from home
14.
Time
15. Casandra runs to her neighbor’s house for a cup of sugar and then runs home.
Sketch a graph of her trip. Graph time on the x-axis and distance from home
on the y-axis.
16. A monkey climbed a tree at a constant rate and then jumped to the ground.
Sketch a graph of his adventure. Graph time on the x-axis and speed on the y-axis.
17. Manny walked to his friend’s house, stayed to play video games and then walked
home. The graph below shows his distance from home (feet) over time (minutes).
(95, 1000)
Distance from home
(feet)
(5, 1000)
(105, 0)
(0, 0)
Time (minutes)
162
Block 4 ~ Review
a. How far away does Manny’s friend live?
b. How long did Manny stay at his friend’s house?
c. Find Manny’s rate of speed in feet per minute for the
trip to his friend’s house.
d. Find Manny’s rate of speed in feet per minute for
the trip home from his friend’s house.
Lesson 22 ~ Direct Variation Tables and Graphs
each table below represents a direct variation relationship. graph each scatter plot. Find the rate by
which each graph increases.
18. x
y
0
0
1
1
2
2
3
3
4
4
19. x
0
y
20. x
0
y
0
0
0.5
1
2
1
2
4
2
1
3
6
3
1.5
4
8
4
2
determine whether the table models direct variation. explain why or why not. If it does, give the rate.
21. x
y
22. x
y
0
0
1
23. x
y
0
0
0
0
2
1
4
1
1
2
4
3
6
2
8
2
2
3
10
3
3
4
8
4
12
4
4
determine whether the graph models direct variation. explain why or why not. If it does, give the rate.
24.
25.
(3, 9)
(4, 5)
(1, 3)
(1, 2)
Lesson 23 ~ Direct Variation Equations
Copy and complete each input-output table and graph each function in Quadrant I.
26.
Input
x
Function rule
y = 3x
output
y
27.
Input
x
0
0
1
1
2
2
3
3
4
4
Function rule
y=x
output
y
Block 4 ~ Review
163
28.
Input
x
Function rule
y = __23 x
0
29.
output
y
Input
x
Function rule
y = 0.4x
output
y
0
1
3
2
6
3
9
4
12
Find the slope of each direct variation equation.
30. y = 2x
32. y = _13 x
31. y = x
Write a direct variation equation for each table.
33. x
y
34. x
y
35. x
y
0
0
0
0
0
0
1
9
1
0.25
2
30
2
18
2
0.5
4
60
3
27
3
0.75
5
75
Lesson 24 ~ Recognizing Direct Variation
tell whether or not each graph is a direct variation graph. explain your choice.
36.
37.
38.
tell whether or not each equation is a direct variation equation. explain your choice. If it is direct
variation, identify the slope.
39. y = 10x + 4
40. y = 2.5x
41. y = − _12 x
42. y = _2x
43. y = −4x²
44. y = _25 x
tell whether or not each table shows ordered pairs that model direct variation. explain your choice. If the
ordered pairs show direct variation, identify the slope.
45. x
y
46. x
y
47. x
y
−2
−8
−2
2
−2
−1
−1
−4
−1
1
−1
−2
0
0
1
−1
1
2
1
4
2
−2
2
1
2
8
−3
3
1.5
164
Block 4 ~ Review
3
graph each direct variation equation on a four quadrant coordinate plane.
48. y = 2x
49. y = 0.25x
50. y = −3x
51. Leonard pays his son $5.00 an hour to rake leaves in the fall.
a. Copy and complete the table below. The x-value is the
number of hours Leonard’s son rakes leaves and
the y-value is the amount of money he makes.
hours, x
0
1
2
3
4
5
6
Money earned, y
b. Does the table in part a show direct variation?
Explain why or why not.
c. If the table shows direct variation, find the slope of the
direct variation function.
d. If the table shows direct variation, write the direct variation
equation y = mx and replace the m with the slope from part c.
Lesson 25 ~ Writing Linear Equations
Find the linear equation for each input-output table or graph.
52.
54.
53.
Input
x
output
y
−2
15
−1
12
0
9
1
6
2
3
3
0
Input
x
output
y
−2
−6
−1
−1
0
4
1
9
2
14
3
19
55.
Block 4 ~ Review
165
56. Penny opened a savings account with $60. Each month she
added $15 to the account.
a. Copy and complete the input-output table based on
the information above.
b. Find the linear equation for the table.
c. Use the linear equation to determine her account
balance after one year.
57. Find the missing values in the table below and write the
Input
Months, x
output
Account Balance, y
0
1
2
3
4
5
linear equation for the table.
Input
x
output
y
−2
−5
−1
−1
0
7
2
t ic -t Ac -t oe ~ B e A n B Ag t o s s
Elena and Kelly are making a bean bag toss game for the school carnival.
Elena wants to cut two rectangles out of the plywood for the holes.
Kelly put a coordinate plane on the piece of plywood. She labeled the x-axis from
–10 to 10 and the y-axis from –10 to 10. Next, she picked four points for the first
rectangular hole and four more points for the next rectangular hole.
step 1: Graph each of Kelly’s rectangles.
Blue ~ (−2, 2), (8, 2), (8, 7), (−2, 7)
Pink ~ (−8, −5), (8, −5), (8, −2), (−8, −2)
step 2: Find the area of each of the rectangles.
step 3: Find the area of the entire piece of plywood.
area of blue and pink rectangles
step 4: Find the percent of the board cut out by the rectangles. Find ______________________.
area of plywood
step 5: Their teacher actually wants the rectangles to take up between 40% and 50% of the plywood.
Find new coordinates for new blue and pink rectangles that fit this requirement. Show the
rectangles work by graphing them and repeating the calculations from steps 1, 2 and 4.
166
Block 4 ~ Review
steVe
AutomotiVe techniciAn
clAckAmAs, oregon
CAreer
FoCus
I am an automotive technician. I repair and service cars. When a customer
brings his car to me to service, I road test the vehicle and do a thorough
inspection. This helps me determine what work needs to be done and what
I need to service so the car will run smoothly. If a customer brings me a vehicle
that needs a repair, I try to figure out what is wrong and why it is not working properly.
After I figure out what needs to be done, I give the customer a price quote on how much the repair
will cost. If the customer agrees with the price, I make the repairs. I test the car after repairing it to
make sure it is running well.
Auto technicians use math on a daily basis. I measure and use percentages to determine if brakes are
still safe for a car. I check measurements on different parts to the thousandths of an inch. Cars are
complex machines. Everything needs to work well for the car to run. If something is worn down even
a little bit, it can cause the car to break down. Math helps me determine when a part is too worn and
needs to be replaced.
The best education for becoming an auto technician is experience. However, a two year degree from
vocational school makes a good start. New cars come out every year with better technology and different
parts. Auto technicians must continue to get training on a constant basis to keep up with the changes.
An average starting salary for a full time automotive technician is about $35,000 per year. After several
years of experience, a technician can expect to make $45,000-$90,000 per year. Much of a technician’s
salary depends on experience, skill level and ability.
When I was growing up, I loved taking things apart, figuring out how they worked and then putting
them back together again. A love for cars and engines led me to work on them as a teenager. It was a
natural fit for me to go into the field of automotive repair. I enjoy taking a vehicle that is not working,
figuring out what is wrong with it and making it right again.
Block 4 ~ Review
167