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

 Page 8 / 19

3

5

3

5

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 ( $x$ ).

#### Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

 By By By Anonymous User