Entry Task – SHOW WORK

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2.2 Direct Variation

Target: I canWrite and interpret direct

variation equations

Definition:

Y varies directly as x means that y = kx where k is the constant of variation.

Another way of writing this is k = In other words:

* the constant of variation (k) in a direct variation is the constant (unchanged) ratio of two variable quantities.

y

x

When we talk about a direct variation, we are talking about a relationship where as x

increases, y increases or decreases at a CONSTANT RATE.

Direct Variation

y1

x1

y2

x2

Direct variation uses the following formula:

Direct Variationexample:

if y varies directly as x and y = 10 as x =2.4, find x when y =15. what x and y go together?

Direct VariationIf y varies directly as x and y = 10

find x when y =15.

y = 10, x = 2.4 make these y1 and x1

y = 15, and x = ? make these y2 and x2

Direct Variation

if y varies directly as x and y = 10 as x = 2.4, find x when y =15

10

2.4

15

x

Direct Variation

How do we solve this? Cross multiply and set equal.

10

2.4

15

x

Direct Variation

We get: 10x = 36

Solve for x by diving both sides by

10.

We get x = 3.6

Direct Variation

Let’s do another.If y varies directly with x and y = 12 when x = 2, find y when x = 8.

Set up your equation.

Direct Variation

If y varies directly with x and y = 12 when x = 2, find y when x = 8.

12

2

y

8

Direct Variation

Cross multiply: 96 = 2ySolve for y. 48 = y.

12

2

y

8

X Y 10 30 5 15 3 9

Note: X decreases,

10, 5, 3

And Y decreases.

30, 15, 9

What is the constant of variation of the table above?

Since y = kx we can say Therefore:

30/10=k or k = 3 15/5=k or k = 3

9/3=k or k =3 Note k stays constant.

y = 3x is the equation!

yk

x

Examples of Direct Variation:

Find the value of k.

y=kx Direct variation formula

42 = k(7) Replace y with 15 and x with 5.

42= k(7) 7 7 Divide each side by 56 = k Simplify

Equation?

y = 6x

Y varies directly as x. Find the constant of variation and write an

equation for the direct variation when y = 42 when x = 7

Tell if the following graph is a Direct Variation or not.

No Yes

No No

No Yes

Yes No

Tell if the following graph is a Direct Variation or not.

1. 2.

3.4.

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

Find y when x = -8. HOW???

2 step processX Y

-5 6

-8 ?

1. Find the constant variation.

k = y/x or k = 6/-5 = -1.2

k = -1.2

2. Use y = kx. Find the unknown (x).

y= -1.2(-8)

x= 9.6

Therefore:

X =-8 when Y=9.6

Using Direct Variation to find unknowns (y = kx)

Using Direct Variation to solve word problems

Problem:

A car uses 8 gallons of gasoline to travel 290 miles. How much gasoline will the car use to travel 400 miles?

Step One: Find points in table

X (gas) Y (miles) 8 290 ? 400

Step Two: Find the constant variation and equation:

k = y/x or k = 290/8 or 36.25

y = 36.25 x

Step Three: Use the equation to find the unknown.400 =36.25x 400 =36.25x36.25 36.25or x = 11.03

Homework

P. 71 # 7-47 odds

Challenge - # 54