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### 2.2 Power Functions with Modeling

Garth Schanock, Robert Watt, Luke Piltz

http://ichiko-wind-griffin.deviantart.com/art/Lame-Math-Joke

What is a Power Function?

f(x)=kx^a

Where:

- k is the constant of variation or constant of proportion
- a is the power
- k and a are not zeros

Examples of Power Functions.

Determine whether the function is a power function. If it is a power function, give the power and constant of variation.

1. f(x)=83x⁴

2. f(x)=13

Solutions:

1. f(x)=83x⁴

Yes, it is a power function because it is in the form f(x)=k*xᵃ. The power, or a, is 4. The constant, or k, is 83.

2. f(x)=13

Yes, this is a power function because it is in the form f(x)=k*xᵃ. The power is 0 and the constant is 13. Because anything to the power of zero is one, there isn't an x with the 13.

Examples of Monomials

Determine whether the function is monomial. If it is, give the power and constant. If it isn't, explain why.

1. f(x)=-7

2. f(x)=3x^(-3)

Solutions-

1. f(x)=-7

Yes, the function is a monomial. The power is 0 and the constant is -7.

2. f(x)=3x^(-3)

No, this function is not a monomial function. It is not a monomial function because the power is not a positive integer.

Even and Odd Functions

f(x)=xⁿ is an even function if n is even

Ex. f(x)=3x⁶ f(x)=4x-⁶

f(x)=xⁿ is an odd function if n is odd

Ex. f(x)=3x⁵ f(x)=82x⁹

Even: f(x)=x⁶

Odd: f(x)=x³

http://www.wmueller.com/precalculus/families/1_41.html

Writing power functions

Write the statement as power function. Use k as the constant of variation if one is not specified.

1. The area A of an equilateral triangle varies directly as the square of the length s of its sides.

2. The force of gravity, F, acting on an object is inversely proportional to the square of the distance, d, from the object to the center of the Earth.

Solutions-

1. A=ks² It begins with the area, A, which varies directly with the square of s, or s². Since no constant of variation was given we use k.

2. F=k/d² It begins with the force of gravity, or F, which is inversely proportional to the square of the distance to the center of the Earth, or d². Because it is inversely proportional, it is the denominator. No constant of variation is given so k is used.

State the following for each function

Domain and Range

Continuous or noncontinuous

Describe graph

Bounded above, below ,or no bound

Extrema

State all asymptotes

End behavior

The Cubic Function

F(x)=x^3

Domain: All reals

Range: All reals

Continuous

Increasing for all x

Not bounded

No local extrema

No Horizontal Asymptotes

No Vertical asymptotes

End behavior: (-∞, ∞)

http://library.thinkquest.org/2647/algebra/ftevenodd.htm

The Square Root Function

F(X)= √x

Domain:[0,∞)

Range: [0,∞)

Continuous on [0,∞)

Increasing on [0,∞)

Bounded below but not above

Local minimum at x=0

No Horizontal asymptotes

No vertical asymptotes

End Behavior: [0,∞)

http://onemathematicalcat.org/Math/Algebra_II_obj/basic_models.htm

Sources

Demana, Franklin D. Precalculus: Graphical, Numerical, Algebraic. Boston: Addison-Wesley, 2007. Print.

All other sources are listed under pictures.

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