# 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.

The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris