# 3.3 Power functions and polynomial functions

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In this section, you will:
• Identify power functions.
• Identify end behavior of power functions.
• Identify polynomial functions.
• Identify the degree and leading coefficient of polynomial functions.

Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in [link] .

 Year $2009$ $2010$ $2011$ $2012$ $2013$ Bird Population $800$ $897$ $992$ $1,083$ $1,169$

The population can be estimated using the function $\text{\hspace{0.17em}}P\left(t\right)=-0.3{t}^{3}+97t+800,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}P\left(t\right)\text{\hspace{0.17em}}$ represents the bird population on the island $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.

## Identifying power functions

In order to better understand the bird problem, we need to understand a specific type of function. A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number. (A number that multiplies a variable raised to an exponent is known as a coefficient.)

As an example, consider functions for area or volume. The function for the area of a circle with radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ is

$A\left(r\right)=\pi {r}^{2}$

and the function for the volume of a sphere with radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ is

$V\left(r\right)=\frac{4}{3}\pi {r}^{3}$

Both of these are examples of power functions because they consist of a coefficient, $\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\frac{4}{3}\pi ,\text{\hspace{0.17em}}$ multiplied by a variable $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ raised to a power.

## Power function

A power function    is a function that can be represented in the form

$f\left(x\right)=k{x}^{p}$

where $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ are real numbers, and $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is known as the coefficient    .

Is $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}\text{\hspace{0.17em}}$ a power function?

No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.

## Identifying power functions

Which of the following functions are power functions?

All of the listed functions are power functions.

The constant and identity functions are power functions because they can be written as $\text{\hspace{0.17em}}f\left(x\right)={x}^{0}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(x\right)={x}^{1}\text{\hspace{0.17em}}$ respectively.

The quadratic and cubic functions are power functions with whole number powers $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(x\right)={x}^{3}.$

The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as $\text{\hspace{0.17em}}f\left(x\right)={x}^{-1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(x\right)={x}^{-2}.$

The square and cube root functions are power functions with fractional powers because they can be written as $\text{\hspace{0.17em}}f\left(x\right)={x}^{1/2}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}f\left(x\right)={x}^{1/3}.$

Which functions are power functions?

$\begin{array}{l}f\left(x\right)=2{x}^{2}\cdot 4{x}^{3}\hfill \\ g\left(x\right)=-{x}^{5}+5{x}^{3}-4x\hfill \\ h\left(x\right)=\frac{2{x}^{5}-1}{3{x}^{2}+4}\hfill \end{array}$

$f\left(x\right)\text{\hspace{0.17em}}$ is a power function because it can be written as $\text{\hspace{0.17em}}f\left(x\right)=8{x}^{5}.\text{\hspace{0.17em}}$ The other functions are not power functions.

## Identifying end behavior of power functions

[link] shows the graphs of $\text{\hspace{0.17em}}f\left(x\right)={x}^{2},\text{\hspace{0.17em}}g\left(x\right)={x}^{4}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}h\left(x\right)={x}^{6},\text{\hspace{0.17em}}$ which are all power functions with even, whole-number powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.

#### Questions & Answers

how fast can i understand functions without much difficulty
Joe Reply
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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