Direct and Inverse

Variations

Do now: if 5 boxes of salt costs 15 dollars, how much does 4 boxes cost?

Today we will be able to solve and graph Inverse

Variation problems

Do now: if 5 boxes of salt costs 15 dollars, how much does 4 boxes cost?

5

15=

4x

x=12

This is a direct variation problem. We set them upAs a proportion and solve.

Direct Variation

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

y1x1

=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=

15x

Direct Variation

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

10

2.4=

15x

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=

y8

Direct Variation

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

12

2=

y8

Inverse Variation

Inverse is very similar to direct, but in an inverse relationship as one value goes up, the other goes down. Can you think of an exampleOf this happening?

Inverse Variation

With Direct variation we Divide our x’s and y’s. In Inverse variation we will Multiply them.

x1y1 = x2y2

Inverse Variation

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

x1y1 = x2y2

2(12) = 8y 24 = 8y y = 3

Inverse Variation

If y varies inversely with x:

X 4 8 2 x

Y 6 3 y -1

Inverse Variation

If y varies inversely with x and allProducts =24

X 4 8 2 X=-24

Y 6 3 Y=12 -1

Inverse VariationIf y varies inversely as x and x = 18 when y = 6, find y when x = 8.

18(6) = 8y 108 = 8y y = 13.5

Graphing inverse variations-the graph is a hyperbola.

xy = c where c is the product of x and y

Ex: xy = 12 then y=

12x

Example:

If it takes 3 carpenters to frame a house in 8 weeks, how many weeks will it take four carpenters to frame the same house?

Example:

If it takes 3 carpenters to frame a house in 8 weeks, how many weeks will it take four carpenters to frame the same house?

(3)(8) = (4)(w)W =6

Exit ticket

Create an inverse variation problem with your partner and hand it in on an index card for 3 bonus points on your next test!