5.2: Direct Variation

Joe shovels snow for his neighbor in Buffalo, NY. Joe’s neighbor typically pays Joe a rate of $5.00 for every inch of snow that falls with each snowstorm. An equation relating

Joe’s pay, P, and the amount of snow that falls, h can be modeled by .

We can say that Joe’s pay is PROPORTIONAL to the height of the snowfall.

In other words, Joe’s pay VARIES DIRECTLY as the height of the snowfall.

5P h

All direct variation relationships can be represented by the equation where k≠0

 

k is the coefficient of x and is called the COEFFICIENT of variation.

 

Key Concept: If y varies DIRECTLY as x,

When x INCREASES, y increases by the same RATE.

When x DECREASES, y decreases by the same RATE.

y kx

Solving for k, the constant can be written as the ratio:

Are the following direct variation relationships?

a) The cafeteria charges $1.50 per slice of pizza

YES; Cost increases as the number of slices increase

b) The temperature and the time of day.

NO; Other factors: Storms, cold fronts, etc

yk

x

 Identifying a Direct Variation: To determine whether an equation represents direct variation, SOLVE it for y. If you can write the equation in the form , it represents direct variation.

Does the equation represent a direct variation? 1) 2.)

y kx

4 6 0x y 2 3 6y x 4 4x x

6 4y x 6 6

2

3y x

2 2

33

2y x

NOYES

Writing a Direct Variation Equation: In order to write a direct variation equation, you must first use an ORDERED pair other than (0, 0) to find k. Ex: Suppose y varies directly as x. When x is 12 and y is -4. Find the constant of variation.

Write a direct variation equation to represent the problem.

What is the values of y when x is 8?

4

12

yk

x

1

3k

1

3y x

18

3y

8

3y

Your turn: y varies directly as x. When y = 40 and x = 8. Find the value of y when x = 12.

40

8

yk

x 5k

5y x 60y 5 12y

Graphing a Direct Variation Equation: Using your graphing calculator: Graph the following direct variation equations:

3

2

4

36 7

y x

y x

y x

x y

What is the same about all the graphs?

What is different?

What is the slope of each line?

They all pass through the origin; y-intercept

Their “steepness”; slope

4 63, 2, ,

3 7

Writing a Direct Variation From a Table: For the data in each table, tell whether y varies directly with x. If it does, write an equation for the direct variation.

1.) Is k constant?

Eq:

2.51.25

2k

8.751.25

7k

6.25

1.255

k

YES

1.25y x

2.) Is k constant?

Eq:

10.81.2

9k

14.41.2

12k

3.61.2

3k

NO

NONE

Homework: 5.2 p. 325-327 #’s 14-30 even, 33-37