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WHAT YOU’LL LEARN:
Finding rates of change from tables and graphs.
AND WHY:
To find rates of change in real-world situations, such as the
rate of descent for a parachute or the cost of renting a
computer.
Suppose you type 140 words in 4 minutes. What is your typing rate?
The number of words depends on the number of minutes you type.
So, the number of words is the dependent variable.
Write each as a rate.
a. You buy 5 yards of fabric for $19.95.
b. You travel 268.8 mi on 12 gal of gasoline.
You use a rate of change to find the amount of one quantity per one unit of another, such as typing 35 words
in 1 min.
The RATE OF CHANGE allows you to see the relationship between two
quantities that are changing.
On a graph, you show the dependent variable on the vertical axis and the
independent variable on the horizontal axis.
The SLOPE of a line is the RATE OF CHANGE relating the variables.
Find the rate of change for data graphed on the line. The explain what the rate of change means in this situation.
(0,2.5)
(40,1.5)
Heig
ht
(thousa
nd f
eet)
Time (seconds)0 20 40 60
3
2
1
Find two points on the graph. Use the points to find the slope, which is also the rate of change.
= 40 – 0
1.5 – 2.5
horizontal change vertical change slope =
= 40
1
The rate of change is , which means that the parachute descends 1000 ft every 40 seconds.
401
Why is the height the dependent variable?
(0,2.5)
(40,1.5)H
eig
ht
(thousa
nd f
eet)
Time (seconds)0 20 40 60
3
2
1
The height of the parachute depends on the time it
has descended.
How many feet does the parachute descend in a
second?
25 ft
# of days
Hraibe’s Rental Company
1 $60
2 $75
3 $90
4 $105
5 $120
You can rent a computer from Hraibe’s Rental Company. The first day’s rent
is $60. Find the rate of change for renting a computer after the first day.
Rate of change =
=
=
= =
The rate of change is 15/1, which means that it costs
$15 for each day a computer is rented after the
first day.
# of days
Hraibe’s Rental Company
1 $60
2 $75
3 $90
4 $105
5 $120
Will the rate of change for the data in the
table be the same for any pair of data
items?
Yes; between any two pair of values in the
table, the rate of change is $15 per
day.
x y
0 13 36 59 7
The graph of a LINEAR FUNCTION is a line. You can also use a table to tell whether a relationship between sets of data is linear.
0 2 4 6 8
x
y 8
6
4
2
2 units
3 units
GraphTable
23
23
23
2 units3 units
As you can see from the table for the graph, the rate of change between consecutive pairs of data is constant. So, the
relationship between the x-values and the y-values is linear.
x y
0 13 36 59 7
0 2 4 6 8
x
y 8
6
4
2
2 units
3 units
GraphTable
23
23
23
2 units3 units
Find the rate of change for each situation.
1. A baby is 18 in. long at birth and 27 in. at ten months.
2. The cost of group tickets for a museum is $48 for four people and $78 for ten people.
3. You drive 30 mi in one hour and 120 mi in four hours.
Cost
(dolla
rs)
Weight (ounces)
0 1 2 3 4 5 6
2
1
Find the rate of change.
$1 buys 4 oz of oregano.
Find the rate of change.
About 1 gal used for every 15 mi.
Fuel in
Tank
( gallo
ns)
Miles Traveled0 100 200 300
15
10
5
Is the relationship shown by the data in the table linear?
YES
x y
3 14 -25 -56 -8
-31
-31
-31
Is the relationship shown by the data in the table linear?
NO
x y
2 04 16 48 9
12
32
52
Is the relationship shown by the data in the table linear?
YES
x y
0 31 62 93 12
31
31
31
Is the relationship shown by the data in the table linear?
NO
x y
-3 5-1 8 112 317
32
42
52
Is the relationship shown by the data in the table linear?
YES
x y
1 0 8 115 222 3
17
17
17
Is the relationship shown by the data in the table linear?
YES
x y
9 -4 5 1 1 6-3 11
5-4
5-4
5-4
Is the relationship shown by the data in the table linear?
YES
x y
1 33 74 9715
42
21
63
Rate of change is 2.
