2.2 Power Functions
with Modeling
Garth Schanock, Robert Watt, Luke Piltz
What is a Power Function?
k is the constant of variation or constant of
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
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
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)=xⁿ is an odd function if n is odd
Ex. f(x)=3x⁵
Even: f(x)=x⁶
Odd: f(x)=x³
Writing power functions
Write the statement as power function. Use k as the constant of variation if
one is not specified.
The area A of an equilateral triangle varies directly as the square of the
length s of its sides.
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.
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.
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
Domain and Range
Continuous or noncontinuous
Describe graph
Bounded above, below ,or no bound
State all asymptotes
End behavior
The Cubic Function
Domain: All reals
Range: All reals
Increasing for all x
Not bounded
No local extrema
No Horizontal Asymptotes
No Vertical asymptotes
End behavior: (-∞, ∞)
The Square Root Function
F(X)= √x
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,∞)
Demana, Franklin D. Precalculus: Graphical, Numerical,
Algebraic. Boston: Addison-Wesley, 2007. Print.
All other sources are listed under pictures.

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