3.3 Power functions and polynomial functions  (Page 8/19)

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

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.

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 ( $x$ ).

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich