Download 2.2 Power Functions with Modeling

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
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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
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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.