Power Functions

Objectives

• Students will: Have a review on converting radicals to

exponential form Learn to identify, graph, and model power

functions

Converting Between Radical and Rational Exponent Notation

• An exponential expression with exponent of the form “m/n” can be converted to radical notation with index of “n”, and vice versa, by either of the following formulas:

1. n mn

m

aa 3

2

8 4643 3 28

Write in radical form.

Write in radical form.

Write each expression in radical form.

a.

b.

Answer:

Answer:

Write using rational exponents.

Answer:

Write using rational exponents.

Answer:

Write each radical using rational exponents.

a.

b.

Answer:

Answer:

Examples

.

7

4

5

5 98 5

9

8

11

3

4x 11 3)4( x

7 45

Power Function

• Definition Where k and p

are non zero constants

• Power functions are seen when dealing with areas and volumes

• Power functions also show up in gravitation (falling bodies)

py k x

34

3v r

216velocity t

Direct Proportions

• The variable y is directly proportional to x when: y = k * x• (k is some constant value)

• Alternatively

• As x gets larger, y must also get larger• keeps the resulting k the same

yk

x

This is a power function

This is a power function

Direct Proportions

• Example: The harder you hit the baseball The farther it travels

• Distance hit is directlyproportional to theforce of the hit

Direct Proportion

• Suppose the constant of proportionality is 4 Then y = 4 * x What does the graph of this function look like?

Inverse Proportion

• The variable y is inversely proportional

to x when

• Alternatively y = k * x -1

• As x gets larger, y must get smaller to keep the resulting k the same

ky

x

Again, this is a power function

Again, this is a power function

Inverse Proportion

• Example:If you bake cookies at a higher temperature, they take less time

• Time is inversely proportional to temperature

Inverse Proportion• Consider what the graph looks like

Let the constant or proportionality k = 4

Then 4

yx

Power Function

• Looking at the definition

• Recall from the chapter on shifting and stretching, what effect the k will have? Vertical stretch or compression

py k x

for k < 1

Power Functions

• Parabola y = x2

• Cubic function y = x3

• Hyperbola y = x-1

Power Functions

• y = x-2

•

•

1

2y x

133y x x

Power Functions

• Most power functions are similar to one of these six

• xp with even powers of p are similar to x2

• xp with negative odd powers of p are similar to x -1

• xp with negative even powers of p are similar to x -2

• Which of the functions have symmetry? What kind of symmetry?

Variations for Different Powers of p

• For large x, large powers of x dominate

x5x4

x3

x2

x

Variations for Different Powers of p

• For 0 < x < 1, small powers of x dominate

x5x4

x3x2

x

Variations for Different Powers of p

• Note asymptotic behavior of y = x -3 is more extreme

y = x -3 approaches x-axis more rapidly

0.5

0.510

20

y = x -3 climbs faster near the y-axis

1

x

2x

2x

1

x

Think About It…

• Given y = x –p for p a positive integer

• What is the domain/range of the function? Does it make a difference if p is odd or even?

• What symmetries are exhibited?

• What happens when x approaches 0

• What happens for large positive/negative values of x?

Finding Values

• Find the values of m, t, and k

4

3( )g x kx

1

3( )f x mx(8,t)

Homework

• Pg. 189 1-49 odd