RATE OF CHANGE AND
DIRECT VARIATION
SECTION 5.3
VERTICAL AXIS
Rate of Change 
change in VERTICAL AXIS
change in HORIZONTAL AXIS
HORIZONTAL
CHANGE
VERTICAL CHANGE
HORIZONTAL AXIS
slope
Rate of change is related to the __________
of
a line because they are both a ratio of the change
vertical over the change in __________.
horizontal
in ______
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.
1600
rise 800miles
speed 

run 80 minutes
80
1200
800
800
Distance (miles)
2000
400
20
40
60
80
Time (minutes)
100
120
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.
160
rise
speed 

run
1.0
120
80
60 miles
1.0 hours
60
Distance (miles)
200
40
0.5
1.0
1.5
2.0
Time (hours)
2.5
3.0
rise 50 feet
speed 

run 3 seconds
3
50
Distance (feet)
100
80
60
40
20
1
2
3
4
Time (seconds)
5
6
rise
speed 

run
2000
4
1500
1000
750
Distance (miles)
2500
500
4
8
12
16
Time (days)
20
24
750 miles
4 days
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.
The student walked forward
5 feet in 3 seconds.
Distance (feet)
10
He then walked backwards 3
feet in 2 seconds.
8
He stood still for 1 second.
6
He then walked forward 2
feet in 3 seconds.
4
2
2
4
6
8
Time (seconds)
10
12
One type of rate of change is ______________.
direct variation
y
k
y  kx
x
If y varies directly as x, then ____________, or _____________,
where k is the constant of variation.
Example 2: Looking For the Constant of Variation (k)
A. If y varies directly as x and y  8 when x  4 , find the constant of variation
and write the equation for direct variation.
y
8
k
k
4
x
y  kx
k2
y  2x
k2
y  2x
B. If y varies directly as x and y  2 when x  5 , find the constant of variation
and write the equation for direct variation.
y  kx
2
y x
5
y
2
k
k
5
x
2
k 
5
2
y x
5
2
k 
5
If y varies directly as x and y  14 when x  2 , find the constant of variation
and write the equation for direct variation.
y
14
k
k
2
x
y  kx
k 7
y  7x
k 7
y  7x
Example 3: Looking For the missing value of x or y.
A. If y varies directly as x and
y  27
y1
when
x  6 , find x when y  45
x1
x2
You can set up and solve a proportion to find x or y.
y1 y2

x1 x2
27 45

6
x
27 x  270
27
27
x  10
.
y2
A. If y varies directly as x and
y1 y2

x1 x2
y  35
when
y1
x  7 , find y when x  84
x1
y2
35 y

7 84
2940  7 y
7
7
y  420
.
x2
If y varies directly as x and
y1 y2

x1 x2
y  36
y1
when
x  9 , find x when y  48
x1
x2
36 48

9
x
36 x  432
36
36
x  12
.
y2