Power Functions with Modeling
DEFINITION: POWER FUNCTION
Any function that can be written in the form
f(x) = k ·x ⁿ, where k and n are nonzero
constants
is a power function. The constant n is the power, and
k is the constant of variation, or constant of
proportion. We say f(x) varies as the nth power of x,
or f(x) is proportional to the nth power of x.
Power function models involve output-form-input
relationships that can be expressed in the language of
variation and proportion:
The power function formulas with positive powers are
statements of direct variation
and power function formulas with negative powers are
statements of inverse variation.
C = 2∏r
The circumference of a circle
varies directly as its radius
A = ∏r²
The area enclosed by a circle is
directly proportional to the
square of its radius
F = k/d²
The force of gravity acting on an object is
inversely proportional to the square of the
distance from the object to the center of the
Earth
V = k/p
The volume of an enclosed gas
varies inversely as the applied
pressure.
Writing a Power Function Formula
• From empirical evidence and the laws of
physics it has been found that the period of
time T for the full swing of a pendulum varies
as the square root of the pendulum’s length l,
provided that the swing is small relative to the
length of the pendulum.
Solution:
• Because it does not state otherwise, the
variation is direct. So the power is positive.
The wording tells us that T is a function of I.
Using k as the constant of variation gives us
T (l )  k l
Five Basic Power Functions
f ( x)  x
f ( x)  x
f ( x)  x
1
f ( x) 
x
3
2
f ( x)  x
If you put any real number
in this function, you get the
same real number “back”.
f(x) = x
Identity Function
f(x) = x, slope 1, y-intercept = 0
The domain of this function is all
real numbers.
The range is also all real numbers
1
f x  
x
Reciprocal Function
The domain of this function is all
NON-ZERO real numbers.
The range is all NON-ZERO real numbers.
Square Root Function
f x   x
The domain of this function is
NON-NEGATIVE real numbers.
The range is NON-NEGATIVE real numbers
Cube Function
f(x) = x3
The domain of this function is all
real numbers.
The range is all real numbers
Square Function
f(x) = x2
The domain of this function is all
real numbers.
The range is all NON-NEGATIVE real numbers