Variation

Variation

Objectives: - Write equations for direct, inverse, and joint variations. - Use variation equations to make predictions.

Consider

How does the number of cookies relate to cost?

How much would zero cookies cost?

Consider

The cost is directly related to the number of cookies.Write an equation to represent the cost for x number of cookies.

Cost (Tenths of Dollars)

Number of Cookies

Constant of Variation

Constant of variation is given by k.k describes how variables relate.

For example: 5 candy bars cost $10. 3 candy bars cost $6. k = $2 for each candy bar. You can write this relationship as y = 2x where y is the cost of x number of candy bars.

Direct Variation

Has the form y = kx

Jade read 20 pages in 60 minutes. How many pages will she read in 140 minutes?

Direct Variation

Has the form y = kx

In the previous example, the total cost varies directly with the number of candy bars. y increases by 2 as x increases by 1. y = 2x

Example: The temperature in an oven varies directly with how long it's been warming up. After 5 minutes, the temperature in the oven was 60 degrees. Find k.

Direct Variation

Has the form y = kx

Y varies directly with x. If y = -88 when x = 11, what is y when x = 50?

Zoom poll

y varies directly with x.y is 5 when x is 15. What is the constant of variation? a. 3 b. 1/3 c. 75

Zoom poll

If y varies directly with x and the constant of variation is 8, what equation represents the relationship between x and y?a. y = 8xb. y = 1/8xc. x = 8y

Consider

It takes 15 hours for 9 men to complete a job. How long should it take 15 men to complete the job?

inverse variation

y varies indirectlywith x

inverse variation

y varies indirectly with x

inverse variation

y varies indirectly with x

For example It takes 2 men 6 days to paint a barn, but it takes 3 men 4 days. So k = 12. The relationship is given by In how many days would 5 men paint the fence?

inverse variation

y varies indirectly with x

y varies inversely with x. When x is 3, y is 2.What is the constant of variation?

Zoom poll

If y varies inversely with x, and the constant of variation is 11, what is the equation that describes this relationship?

Brain Break!

Zoom Poll What are you most looking forward to this weekend? a. Sleeping b. Family/Friends c. Food

joint variation

combined variation

z varies directly with x and y z = kxy

z varies directly with x and inversely with y

Direct variation

inverse variation

y varies inversely with x

y varies directly with x y = kx

Decide which formyou are using based on the info in the question.

variation problems

Step 1

A question will typically name two or three variables and how they relate. Your job is to find k, and write a general equation that can be used to find all values in that relation.

Substitute the given values for each variable and solve for k.

Step 2

Example z varies directly with x and inversely with y. When x = 4 and y = 5, z = 8. What is the value of z when x = 8 and y = 10?

Write a general equation, only replacing k with the value you found.

Step 3

Use your general equation to find a value the question asks.

Step 4

Decide which formyou are using based on the info in the question.

variation problems

Step 1

A question will typically name two or three variables and how they relate. Your job is to find k, and write a general equation that can be used to find all values in that relation.

Substitute the given values for each variable and solve for k.

Step 2

Example z varies jointly with x and y. When x = 12 and y = 10, z = 6. What is the value of z when x = 5 and y = 7?

Write a general equation, only replacing k with the value you found.

Step 3

Use your general equation to find a value the question asks.

Step 4

Zoom poll

y varies directly with x. When x = 12.5, y = 25, what is the value of y when x = 50? a. 2 b. 25 c. 100

Zoom poll

y varies inversely with x. When x = 3, y = 4, what is the value of y when x = 6? a. 2 b. 25 c. 100

Zoom poll

z varies jointly with x and y. When x = 3 and y = 4, z = 48. What is the value of z when x = 6 and y = 2? a. 2 b. 48 c. 100

Zoom poll

z varies directly with x and inversely with y. When x = 3 and y = 2, z = 48. What is the value of z when x = 4 and y = 2? a. 32 b. 48 c. 64

Variation

You should be able to: - Write equations for direct, inverse, and joint variations. - Use variation equations to make predictions.