RATE OF CHANGE AND DIRECT VARIATION
SECTION 5.3
change inRate of Change
change in
Rate of change is related to the __________ of a line because they are both a ratio of the change in ______ over the change in __________.
VERTICAL AXIS
HORIZONTAL AXIS
VE
RT
ICA
L A
XIS
HORIZONTAL AXIS
VERTICAL CHANGE
HORIZONTAL CHANGE
slope
vertical horizontal
The graphs below show the distance that an object travels at a constant speed. Example 1: Find the speed (rate of change) for each object.
400
2000
20
1600
1200
800
6040 80 100 120
Time (minutes)
Dis
tanc
e (m
iles)
80
080
risespeed
run 800miles
80 minutes
10 miles p
er minute
The graphs below show the distance that an object travels at a constant speed. Example 1: Find the speed (rate of change) for each object.
40
200
0.5
160
120
80
1.51.0 2.0 2.5 3.0
Time (hours)
Dis
tanc
e (m
iles)
60
1.0
risespeed
run 60 miles
1.0 hours
60 miles p
er hour
20
100
1
80
60
40
32 4 5 6
Time (seconds)
Dis
tanc
e (f
eet) 50
3
risespeed
run 50 feet
3 seconds
16.7 feet p
er second
500
2500
4
2000
1500
1000
128 16 20 24
Time (days)
Dis
tanc
e (m
iles)
75
0
4
risespeed
run 750 miles
4 days
187.5 miles p
er day
The graph below was made from data collected by a motion detector. In the experiment, a student walked in a straight line away from the motion detector. Describe the movement of the person by identifying the rates of change shown on the graph below.
2
10
2
8
6
4
64 8 10 12
Time (seconds)
Dis
tanc
e (f
eet)
The student walked forward 5 feet in 3 seconds. He then walked backwards 3 feet in 2 seconds.
He stood still for 1 second.
He then walked forward 2 feet in 3 seconds.
One type of rate of change is ______________.direct variation
If y varies directly as x, then ____________, or _____________, where k is the constant of variation.
y kxy
kx
Example 2: Looking For the Constant of Variation (k)
A. If y varies directly as x and 8y when 4x , find the constant of variation
and write the equation for direct variation.
y kxk 8
4y
kx
2k
y kx
2y x 2k 2y x
B. If y varies directly as x and 2y when 5x , find the constant of variation
and write the equation for direct variation.
y kxk 2
5y
kx
2
5k
y kx
2
5y x
2
5k
2
5y x
If y varies directly as x and 14y when 2x , find the constant of variation
and write the equation for direct variation.
y kxk 14
2y
kx
7k
y kx
7y x 7k 7y x
Example 3: Looking For the missing value of x or y.
A. If y varies directly as x and 27y when 6x , find x when 45y
.
You can set up and solve a proportion to find x or y.
1 2
1 2
y y
x x 27 45
6 x
1y 1x 2x 2y
27 270x 27 27
10x
A. If y varies directly as x and 35y when 7x , find y when 84x
.
1 2
1 2
y y
x x 35
7 84
y
1y 1x 2x2y
2940 7y7 7
420y
If y varies directly as x and 36y when 9x , find x when 48y
.
1 2
1 2
y y
x x 36 48
9 x
1y 1x 2x 2y
36 432x 36 36
12x