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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

http://brownsharpie.courtneygibbons.org/?p=557

**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*x ᵃ . 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*x power of zero is one, there isn't an x with the 13. ᵃ . The power is 0 and the constant is 13. Because anything to the

**Monomial Functions**

f(x)=k or f(x)=k*x^n Where: • *k *is a constant • *n *is a positive integer

**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) =x ³ 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=*ks*² It begins with the area, A, which varies directly with the square of *s*, or *s*². Since no constant of variation was given we use *k*.

2. *F*=k/*d*² 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 * *d*². 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 Increasing on [0, ∞) [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_mo dels.htm

**Sources**

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

All other sources are listed under pictures.