# 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

"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
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
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
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
divide simplify each answer 3/2÷5/4
divide simplify each answer 25/3÷5/12
Momo