# 5.2 Power functions and polynomial functions  (Page 8/19)

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For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.

Yes. Number of turning points is 2. Least possible degree is 3.

Yes. Number of turning points is 1. Least possible degree is 2.

Yes. Number of turning points is 0. Least possible degree is 1.

No.

Yes. Number of turning points is 0. Least possible degree is 1.

## Numeric

For the following exercises, make a table to confirm the end behavior of the function.

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

$f\left(x\right)={x}^{4}-5{x}^{2}$

$x$ $f\left(x\right)$
10 9,500
100 99,950,000
–10 9,500
–100 99,950,000

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

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

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

$x$ $f\left(x\right)$
10 –504
100 –941,094
–10 1,716
–100 1,061,106

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

$f\left(x\right)=\frac{{x}^{5}}{10}-{x}^{4}$

## Technology

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

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

The $\text{\hspace{0.17em}}y\text{-}$ intercept is The $\text{\hspace{0.17em}}x\text{-}$ intercepts are $\text{As}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to \infty$

$f\left(x\right)=x\left(x-3\right)\left(x+3\right)$

$f\left(x\right)=x\left(14-2x\right)\left(10-2x\right)$

The $\text{\hspace{0.17em}}y\text{-}$ intercept is $\text{\hspace{0.17em}}\left(0,0\right)$ . The $\text{\hspace{0.17em}}x\text{-}$ intercepts are $\text{As}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to -\infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to \infty$

$f\left(x\right)=x\left(14-2x\right){\left(10-2x\right)}^{2}$

$f\left(x\right)={x}^{3}-16x$

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

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

$f\left(x\right)={x}^{4}-81$

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

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

$f\left(x\right)={x}^{3}-2{x}^{2}-15x$

The $\text{\hspace{0.17em}}y\text{-}$ intercept is The $\text{\hspace{0.17em}}x\text{-}$ intercepts are $\text{As}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to -\infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to \infty$

$f\left(x\right)={x}^{3}-0.01x$

## Extensions

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

The $\text{\hspace{0.17em}}y-$ intercept is $\text{\hspace{0.17em}}\left(0,-4\right).\text{\hspace{0.17em}}$ The $\text{\hspace{0.17em}}x-$ intercepts are $\text{\hspace{0.17em}}\left(-2,0\right),\text{\hspace{0.17em}}\left(2,0\right).\text{\hspace{0.17em}}$ Degree is 2.

End behavior: $\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to \infty .$

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

The $\text{\hspace{0.17em}}y-$ intercept is $\text{\hspace{0.17em}}\left(0,9\right).\text{\hspace{0.17em}}$ The $\text{\hspace{0.17em}}x\text{-}$ intercepts are $\text{\hspace{0.17em}}\left(-3,0\right),\text{\hspace{0.17em}}\left(3,0\right).\text{\hspace{0.17em}}$ Degree is 2.

End behavior: $\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to -\infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty .$

The $\text{\hspace{0.17em}}y-$ intercept is $\text{\hspace{0.17em}}\left(0,0\right).\text{\hspace{0.17em}}$ The $\text{\hspace{0.17em}}x-$ intercepts are $\text{\hspace{0.17em}}\left(0,0\right),\text{\hspace{0.17em}}\left(2,0\right).\text{\hspace{0.17em}}$ Degree is 3.

End behavior: $\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to -\infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to \infty .$

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

The $\text{\hspace{0.17em}}y-$ intercept is $\text{\hspace{0.17em}}\left(0,1\right).\text{\hspace{0.17em}}$ The $\text{\hspace{0.17em}}x-$ intercept is $\text{\hspace{0.17em}}\left(1,0\right).\text{\hspace{0.17em}}$ Degree is 3.

End behavior: $\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty .$

The $\text{\hspace{0.17em}}y-$ intercept is $\text{\hspace{0.17em}}\left(0,1\right).\text{\hspace{0.17em}}$ There is no $\text{\hspace{0.17em}}x-$ intercept. Degree is 4.

End behavior: $\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to \infty .$

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

## Real-world applications

For the following exercises, use the written statements to construct a polynomial function that represents the required information.

An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of $\text{\hspace{0.17em}}d,\text{\hspace{0.17em}}$ the number of days elapsed.

A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of $\text{\hspace{0.17em}}m,\text{\hspace{0.17em}}$ the number of minutes elapsed.

$V\left(m\right)=8{m}^{3}+36{m}^{2}+54m+27$

A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ inches and the width increased by twice that amount, express the area of the rectangle as a function of $\text{\hspace{0.17em}}x.$

An open box is to be constructed by cutting out square corners of $\text{\hspace{0.17em}}x-$ inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of $\text{\hspace{0.17em}}x.$

$V\left(x\right)=4{x}^{3}-32{x}^{2}+64x$

A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width ( $x$ ).

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
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Sherica
im all ears I need to learn
Sherica
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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+_
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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×