# 5.2 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

Before we can understand the bird problem, it will be helpful to understand a different 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.

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 $\text{\hspace{0.17em}}f\left(x\right)={x}^{\frac{1}{2}}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}f\left(x\right)={x}^{\frac{1}{3}}.$

Which functions are power functions?

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

$f\left(x\right)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}h\left(x\right)={x}^{6},\text{\hspace{0.17em}}$ 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.

Cos45/sec30+cosec30=
Cos 45 = 1/ √ 2 sec 30 = 2/√3 cosec 30 = 2. =1/√2 / 2/√3+2 =1/√2/2+2√3/√3 =1/√2*√3/2+2√3 =√3/√2(2+2√3) =√3/2√2+2√6 --------- (1) =√3 (2√6-2√2)/((2√6)+2√2))(2√6-2√2) =2√3(√6-√2)/(2√6)²-(2√2)² =2√3(√6-√2)/24-8 =2√3(√6-√2)/16 =√18-√16/8 =3√2-√6/8 ----------(2)
exercise 1.2 solution b....isnt it lacking
I dnt get dis work well
what is one-to-one function
what is the procedure in solving quadratic equetion at least 6?
Almighty formula or by factorization...or by graphical analysis
Damian
I need to learn this trigonometry from A level.. can anyone help here?
yes am hia
Miiro
tanh2x =2tanhx/1+tanh^2x
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation
f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb
favour
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
i am in
Cliff
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
helo
Akash
hlo
Akash
Hello
Hudheifa
which of these functions is not uniformly continuous on 0,1