Over Lesson 8–4
Find the rate of change for the linear
function represented in the graph.
A ski lift goes from the base of a mountain to a
A. A
point near the top of the mountain. After 3 minutes,
the lift had traveled 200 feet. After 6 minutes B.
it had
B
traveled 400 feet. Find the rate of change.
C. C
D. D
You have already identified proportional and
nonproportional relationships in tables and
graphs. (Lesson 6–4)
• Identify proportional and nonproportional
relationships by finding a constant rate of
change.
• Solve problems involving direct variation.
• linear relationship
Relationships that have straight line graphs
• constant rate of change
• direct variation
The rate of change between any
two data points in a linear
relationship is the same or
constant
A special type of linear equation that describes
rate of change. A relationship such that as x
increases in value, y increases or decreases at a
constant rate.
• constant of variation
The rate of change in an equation
y = kx, represented by k
Find a Constant Rate of Change
SAVINGS The graph shows the
amount of money in Yen’s
savings account each week. Find
the constant rate of change. Then
interpret its meaning.
Step 1
Choose any two points on
the line, such as (1, 20)
and (4, 80).
(1, 20) → 1 week, $20
(4, 80) → 4 weeks, $80
Find a Constant Rate of Change
Step 2
Find the rate of change between the
points.
Answer: Yen’s savings account is increasing at a rate of
$20 per week.
The graph shows the amount of
money in Naomi’s savings
account each week. Which
statement describes the rate of
change?
A. increasing at a rate of $15/week
B. decreasing at a rate of $15/week
C. increasing at a rate of $30/week
D. decreasing at a rate of $30/week
A.
B.
C.
D.
A
B
C
D
Use Graphs to Identify Proportional Linear
Relationships
CYCLING The graph shows distances that a cyclist
rides. Determine if there is a proportional linear
relationship between the time and distance.
Use Graphs to Identify Proportional Linear
Relationships
Answer: The ratios are different, so this is not a
proportional relationship.
The graph shows a baby’s
weight gain. Determine if there
is a proportional relationship
between time and weight.
A. Yes, the ratio of weight to
weeks is always the same.
B. Yes, the ratio of weight to
weeks is not always the same.
C. No, the ratio of weight to
weeks is always the same.
D. No, the ratio of weight to
weeks is not always the same.
A.
B.
C.
D.
A
B
C
D
Use Direct Variation to Solve Problems
A. LANDSCAPING As it is being
dug, the depth of a hole for a
backyard pond is recorded in a
table. Write an equation that
relates time and hole depth.
Step 1
Find the value of k using
the equation y = kx.
Choose any point in the
table. Then solve for k.
y = kx
8 = k(10)
Direct variation equation
Replace y with 8 and x with 10.
Use Direct Variation to Solve Problems
Divide each side by 10.
Simplify.
Step 2
Use k to write an equation.
y = kx
Direct variation
y = 0.8x
Replace k with 0.8.
Answer: y = 0.8x
Use Direct Variation to Solve Problems
B. LANDSCAPING As it is being
dug, the depth of a hole for a
backyard pond is recorded in a
table. Predict how long it will take
to dig a depth of 36 inches.
y = 0.8x
36 = 0.8x
Write the direct variation
equation.
Replace y with 36.
Divide each side by 0.8.
Simplify.
Answer: 45 minutes
A. BUSINESS The graph shows
the number of frequent customer
points a bookstore customer
receives for each dollar spent in
the store. Write an equation that
relates the spending s and the
points p.
A.
p = 5s
B.
p = 2s
C.
s = 5p
D.
s = 2p
A.
B.
C.
D.
A
B
C
D
B. BUSINESS The graph shows
the number of frequent customer
points a bookstore customer
receives for each dollar spent in
the store. Predict how many
points a customer receives for a
purchase of $34.40.
A.
164
B.
172
C.
204
D.
220
A.
B.
C.
D.
A
B
C
D