# 5.2 Power functions and polynomial functions  (Page 7/19)

 Page 7 / 19

## Key equations

 general form of a polynomial function $f\left(x\right)={a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$

## Key concepts

• A power function is a variable base raised to a number power. See [link] .
• The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
• The end behavior depends on whether the power is even or odd. See [link] and [link] .
• A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See [link] .
• The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See [link] .
• The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See [link] and [link] .
• A polynomial of degree $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ will have at most $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ x- intercepts and at most $\text{\hspace{0.17em}}n-1\text{\hspace{0.17em}}$ turning points. See [link] , [link] , [link] , [link] , and [link] .

## Verbal

Explain the difference between the coefficient of a power function and its degree.

The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.

As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ decreases without bound, so does $\text{\hspace{0.17em}}f\left(x\right).\text{\hspace{0.17em}}$ As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ increases without bound, so does $\text{\hspace{0.17em}}f\left(x\right).$

What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?

What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As $\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty \text{\hspace{0.17em}}$ and as $\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty .\text{\hspace{0.17em}}$

The polynomial function is of even degree and leading coefficient is negative.

## Algebraic

For the following exercises, identify the function as a power function, a polynomial function, or neither.

$f\left(x\right)={x}^{5}$

$f\left(x\right)={\left({x}^{2}\right)}^{3}$

Power function

$f\left(x\right)=x-{x}^{4}$

$f\left(x\right)=\frac{{x}^{2}}{{x}^{2}-1}$

Neither

$f\left(x\right)=2x\left(x+2\right){\left(x-1\right)}^{2}$

$f\left(x\right)={3}^{x+1}$

Neither

For the following exercises, find the degree and leading coefficient for the given polynomial.

$-3x{}^{4}$

$7-2{x}^{2}$

Degree = 2, Coefficient = –2

$x\left(4-{x}^{2}\right)\left(2x+1\right)$

Degree =4, Coefficient = –2

${x}^{2}{\left(2x-3\right)}^{2}$

For the following exercises, determine the end behavior of the functions.

$f\left(x\right)={x}^{4}$

$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}f\left(x\right)\to \infty$

$f\left(x\right)={x}^{3}$

$f\left(x\right)=-{x}^{4}$

$\text{As}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to -\infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty$

$f\left(x\right)=-{x}^{9}$

$\text{As}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to -\infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty$

$f\left(x\right)=3{x}^{2}+x-2$

$f\left(x\right)={x}^{2}\left(2{x}^{3}-x+1\right)$

$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty$

$f\left(x\right)={\left(2-x\right)}^{7}$

For the following exercises, find the intercepts of the functions.

$f\left(t\right)=2\left(t-1\right)\left(t+2\right)\left(t-3\right)$

y -intercept is $\text{\hspace{0.17em}}\left(0,12\right),\text{\hspace{0.17em}}$ t -intercepts are

$g\left(n\right)=-2\left(3n-1\right)\left(2n+1\right)$

$f\left(x\right)={x}^{4}-16$

y -intercept is $\text{\hspace{0.17em}}\left(0,-16\right).\text{\hspace{0.17em}}$ x -intercepts are $\text{\hspace{0.17em}}\left(2,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-2,0\right).$

$f\left(x\right)={x}^{3}+27$

$f\left(x\right)=x\left({x}^{2}-2x-8\right)$

y -intercept is $\text{\hspace{0.17em}}\left(0,0\right).\text{\hspace{0.17em}}$ x -intercepts are $\text{\hspace{0.17em}}\left(0,0\right),\left(4,0\right),\text{\hspace{0.17em}}$ and

$f\left(x\right)=\left(x+3\right)\left(4{x}^{2}-1\right)$

## Graphical

For the following exercises, determine the least possible degree of the polynomial function shown.

#### Questions & Answers

bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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