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*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. 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) Solutions1. 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. Solutions1. 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_mo dels.htm Sources Demana, Franklin D. Precalculus: Graphical, Numerical, Algebraic. Boston: Addison-Wesley, 2007. Print. All other sources are listed under pictures.