# 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 ( $x$ ).

bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
Joeval
(x2-2x+8)-4(x2-3x+5)
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
master
Soo sorry (5±Root11* i)/3
master
Mukhtar
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
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
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?
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
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
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
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
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
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Miranda
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Miranda
Jeffrey
Jeffrey
Miranda
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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?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey