Date: 2.2 Power Functions with Modeling

Definition Power FunctionAny function that can be written in the form:

is a power function. The constant a is the power, and k is the constant of variation or constant of proportion. We say f(x) varies (directly) as the ath power of x or f(x) is proportional to the ath

power of x.

A power function is also called a monomial function.

f (x) k xa

Graphs of Power Functions

Solving Variation Problems

1. Write an equation that describes the given English statement.

2. Substitute the given pair of values into the equation in step 1 to find k.

3. Substitute k into the equation in step 1.

4. Use the equation from step 3 to answer the problem's question.

Text ExampleThe amount of garbage, G, varies directly as the population, P. Allegheny County, Pennsylvania, has a population of 1.3 million and creates 26 million pounds of garbage each week. Find the weekly garbage produced by New York City with a population of 7.3 million. Step 1 Write an equation:

Step 2 Use the given values to find k.

Substitute 26 for G and 1.3 for P in the direct variation equation. Then solve for k.

26 = 1.3 k26/1.3 = k 20 = k

y varies directly as x is expressed as y kx

By changing letters, write an equation that describes the following English statement: Garbage production, G, varies directly as the population, P.

G kP

The amount of garbage, G, varies directly as the population, P. Allegheny County, Pennsylvania, has a population of 1.3 million and creates 26 million pounds of garbage each week. Find the weekly garbage produced by New York City with a population of 7.3 million.

Step 3 Substitute the value of k into the equation.

Text Example cont.

Step 4 Answer the problem's question. substitute 7.3 for P in G 20P and solve for G.

G = 20P Use the equation from step 3.

G = 20(7.3) Substitute 7.3 for P.

G = 146The weekly garbage produced by New York City weighs approximately 146 million pounds.

G kP Use the equation from step 1.

G 20P Replace k, the constant of variation, with 20.

Inverse Variation

y k/xwhere y varies inversely as x.

k is called the constant of variation.

Describe in words the variation shown by the given equation:

Q

kTH

H varies directly as T and inversely as Q

Joint Variationis a variation in which a variable varies directly as the product of two or more other variables.

y kxz

is read y varies jointly as x and z.

Describe in words the variation shown by the given equation:

M kNP

Q2M varies jointly as N and Pand inversely as the square of Q

ExampleAn object’s weight on the moon, M, varies directly as its weight on Earth, E. A person who weighs 75 kilograms on Earth weighs 12 kilograms on the moon. What is the moon weight of a person who weighs 80 kilograms on Earth?Solution

M = 0.16EM = kE

12 = k•7575 75

0.16 = k

M = 0.16 • 80M = 12.8 kilograms

The moon weight of a person who weighs 80 kilograms on Earth is 12.8 kilograms.

Complete Student CheckpointThe number of minutes needed to solve an exercise set of variation problems varies directly as the number of problems and inversely as the number of people working to solve the problems. It takes 4 people 32 minutes to solve 16 problems. How many minutes will it take 8 people to solve 24 problems?

M kPW

32 k16

4

32 4k

8 k

M 8P

WM

8(24)

8M 24

It will take 8 people 24 minutes to solve 24 problems.

Complete Student CheckpointThe volume of a cone, V, varies jointly as its height, h, and the square of its radius, r. A cone with a radius measuring 5 feet and a height measuring 10 feet has a volume of 120π cubic feet. Find the volume of a cone having a radius of 12 feet and a height of 2 feet.

V khr2

120 ft 3 k(10 ft)(5 ft)2

120 ft 3 250 ft 3k

12

25 k

V 12

25hr2

V 12

25 (2 ft)(12 ft)2

V 1386

25 ft 3

A cone having a radius of 12 feet and a height of 2 feet will have a volume of 434.3 ft3.

or 434.3 ft 3

Find the power regression model for the data:

P(a) 0.20a1.5 or P(a) 0.20a3

2 0.20 a3

Power Functions

With Modeling