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A variation is a relationship between two variables

Now, go to the class site and

take the mini quiz on direct variation

Now go to the class site and take the

inverse variation mini quiz

Examples

Not

Direct Variations

Graphs of Direct Variation

Since y = kx, we can think of a direct variation as

an equation in slope intercept form, with no "b"

y = kx y = mx k = x

So, every direct variation is a straight line with

a y-intercept of (0,0)

The line must pass through the origin

Example 3

Given that y varies directly with x and y = 6 when x = -5, what is y when x = -8

First, make a chart:

x y

-5 6

-8 y

Now, since we have a direct

variation, we know

6 = k(-5)

-6/5 = k

Plug -8 into our formula now

and we get:

y = -6/5(-8)

y = -48/5

Example 1

The following table shows a direct variation. Find the value of k

Since this is a direct variation,

we can plug the numbers into

the formula and solve for k:

y = kx

12 = k(6)

2 = k

x y

6 12

7 14

10 20

Direct Variation

Since this value of k works for all of our pairs,

2 is the constant of variation

Example 4

A direct variation follows the relationship

y = kx

Where k is a constant number, called the

constant of variation

A car uses 8 gallons of gas to drive 200 miles, how much gas will the car use to go 425 miles?

First, make a chart:

Now, since we have a direct

variation, we know

290 = k(8)

25 = k

Example 2

gas miles

8 200

x 425

When this relationship exists, we say

"y varies directly as x"

Plug 425 into our formula now

and we get:

425 = 25x

17 = x

Does the following table show a direct variation?

In order for the table to be a direct variation,

k must be the same for all of the coordinate pairs

So, the car would use 17 gallons of gas for 425 miles

example: y = 3x

Since k is the same for all

4 pairs, this is a direct variation

and the constant of variation

is 3/2

x y

4 6

8 12

12 18

16 24

(4,6)

6 = k(4)

6/4 = 3/2 = k

(6,12)

12 = k(8)

12/8 = 3/2 = k

(12,18)

18 = k(12)

18/12 = 3/2 = k

(16,24)

24 = k(16)

24/16 = 3/2 = k

Graphs of Inverse Variation

Example 3

Given that y varies inversely with x and y = 2 when x = 8, what is x when y = 4

Example 1

First, make a chart:

The following table shows a inverse variation. Find the value of k

Now, since we have an inverse

variation, we know

2 = x / 8

16 = x

x y

8 2

x 4

Plug 4 into our formula now

and we get:

4 = 16 / x

4x = 16

x = 4

Direct and Inverse

Since this is a inverse variation,

we can plug the numbers into

the formula and solve for k:

y = k / x

4 = k / 10

40 = k

x y

4 10

5 8

2 20

Since y = k / x, we cannot have 0 for our x value.

Therefore, the graph can never touch the x-axis

Also, the graph never touches the y-axis.

Every inverse variation graph looks similar:

Inverse Variation

Since this value of k works for all of our pairs,

40 is the constant of variation

An inverse variation follows the relationship

y = k/x

Where k is a constant number, called the

constant of variation

When this relationship exists, we say

"y varies inversely as x"

Example 4

Example 2

It costs $22.50 per person to rent a bus for 20 passengers. How much would it cost per

person for 36 passengers to rent the bus?

Does the following table show an inverse variation?

example: y = 3/x

In order for the table to be an inverse variation,

k must be the same for all of the coordinate pairs

First, make a chart:

Passengers cost

8 22.5

36 y

Now, since we have an inverse

variation, we know

22.5 = k / 8

180 = k

S0, the bus costs $180 altogether

Since k is the same for all

4 pairs, this is an inverse variation and the constant of variation is 100

x y

4 25

5 20

10 10

50 2

Plug 180 into our formula now

and we get:

y = 180 / 36

y = 5

(10,10)

10 = k / 10

100 = k

(50,2)

2 = k / 50

100 = k

(4,25)

25 = k / 4

100 = k

(5,20)

20 = k / 5

100 = k

So, if 36 people rent the bus, it would cost them each $5 per person

Variation

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