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