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

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## 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.

An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
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salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×