# 5.3 Graphs of polynomial functions  (Page 9/13)

 Page 9 / 13

$f\left(x\right)={x}^{5}-2x,\text{\hspace{0.17em}}$ between $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=2.$

$f\left(x\right)=-{x}^{4}+4,\text{\hspace{0.17em}}$ between $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=3$ .

$f\left(1\right)=3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(3\right)=–77.\text{\hspace{0.17em}}$ Sign change confirms.

$f\left(x\right)=-2{x}^{3}-x,\text{\hspace{0.17em}}$ between $\text{\hspace{0.17em}}x=–1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=1.$

$f\left(x\right)={x}^{3}-100x+2,\text{\hspace{0.17em}}$ between $\text{\hspace{0.17em}}x=0.01\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=0.1$

$f\left(0.01\right)=1.000001\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(0.1\right)=–7.999.\text{\hspace{0.17em}}$ Sign change confirms.

For the following exercises, find the zeros and give the multiplicity of each.

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

$f\left(x\right)={x}^{2}{\left(2x+3\right)}^{5}{\left(x-4\right)}^{2}$

0 with multiplicity 2, $\text{\hspace{0.17em}}-\frac{3}{2}\text{\hspace{0.17em}}$ with multiplicity 5, 4 with multiplicity 2

$f\left(x\right)={x}^{3}{\left(x-1\right)}^{3}\left(x+2\right)$

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

0 with multiplicity 2, –2 with multiplicity 2

$f\left(x\right)={\left(2x+1\right)}^{3}\left(9{x}^{2}-6x+1\right)$

$f\left(x\right)={\left(3x+2\right)}^{5}\left({x}^{2}-10x+25\right)$

$-\frac{2}{3}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}\text{multiplicity}\text{\hspace{0.17em}}5\text{,}\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}\text{multiplicity}\text{\hspace{0.17em}}\text{2}$

$f\left(x\right)=x\left(4{x}^{2}-12x+9\right)\left({x}^{2}+8x+16\right)$

$f\left(x\right)={x}^{6}-{x}^{5}-2{x}^{4}$

$\text{0}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}\text{multiplicity}\text{\hspace{0.17em}}4\text{,}\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}\text{multiplicity}\text{\hspace{0.17em}}1\text{,}\text{\hspace{0.17em}}–\text{1}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}\text{multiplicity}\text{\hspace{0.17em}}1$

$f\left(x\right)=3{x}^{4}+6{x}^{3}+3{x}^{2}$

$f\left(x\right)=4{x}^{5}-12{x}^{4}+9{x}^{3}$

$\frac{3}{2}\text{\hspace{0.17em}}$ with multiplicity 2, 0 with multiplicity 3

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

$f\left(x\right)=4{x}^{4}\left(9{x}^{4}-12{x}^{3}+4{x}^{2}\right)$

$\text{0}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}\text{multiplicity}\text{\hspace{0.17em}}6\text{,}\text{\hspace{0.17em}}\frac{2}{3}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}\text{multiplicity}\text{\hspace{0.17em}}2$

## Graphical

For the following exercises, graph the polynomial functions. Note $\text{\hspace{0.17em}}x\text{-}$ and $\text{\hspace{0.17em}}y\text{-}$ intercepts, multiplicity, and end behavior.

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

$g\left(x\right)=\left(x+4\right){\left(x-1\right)}^{2}$

x -intercepts, $\left(1, 0\right)$ with multiplicity 2, with multiplicity 1, $y\text{-}$ intercept As $\phantom{\rule{0.2em}{0ex}}x\to -\infty ,\phantom{\rule{0.2em}{0ex}}f\left(x\right)\to -\infty ,\text{\hspace{0.17em}}\text{as}\phantom{\rule{0.2em}{0ex}}x\to \infty ,\phantom{\rule{0.2em}{0ex}}f\left(x\right)\to \infty .$

$h\left(x\right)={\left(x-1\right)}^{3}{\left(x+3\right)}^{2}$

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

x -intercepts $\text{\hspace{0.17em}}\left(3,0\right)\text{\hspace{0.17em}}$ with multiplicity 3, $\text{\hspace{0.17em}}\left(2,0\right)\text{\hspace{0.17em}}$ with multiplicity 2, $\text{\hspace{0.17em}}y\text{-}$ intercept $\text{\hspace{0.17em}}\left(0,–108\right).\text{\hspace{0.17em}}$ As $x\to -\infty ,\phantom{\rule{0.2em}{0ex}}f\left(x\right)\to -\infty ,\phantom{\rule{0.2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}x\to \infty ,\phantom{\rule{0.2em}{0ex}}f\left(x\right)\to \infty .$

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

$n\left(x\right)=-3x\left(x+2\right)\left(x-4\right)$

x -intercepts with multiplicity 1, $\text{\hspace{0.17em}}y\text{-}$ intercept As $x\to -\infty ,\phantom{\rule{0.2em}{0ex}}f\left(x\right)\to \infty ,\phantom{\rule{0.2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}x\to \infty ,\phantom{\rule{0.2em}{0ex}}f\left(x\right)\to -\infty .$

For the following exercises, use the graphs to write the formula for a polynomial function of least degree.

$f\left(x\right)=-\frac{2}{9}\left(x-3\right)\left(x+1\right)\left(x+3\right)$

$f\left(x\right)=\frac{1}{4}{\left(x+2\right)}^{2}\left(x-3\right)$

For the following exercises, use the graph to identify zeros and multiplicity.

–4, –2, 1, 3 with multiplicity 1

–2, 3 each with multiplicity 2

For the following exercises, use the given information about the polynomial graph to write the equation.

Degree 3. Zeros at $\text{\hspace{0.17em}}x=–2,$ $\text{\hspace{0.17em}}x=1,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=3.\text{\hspace{0.17em}}$ y -intercept at $\text{\hspace{0.17em}}\left(0,–4\right).$

$f\left(x\right)=-\frac{2}{3}\left(x+2\right)\left(x-1\right)\left(x-3\right)$

Degree 3. Zeros at $\text{\hspace{0.17em}}x=\text{–5,}$ $\text{\hspace{0.17em}}x=–2,$ and $\text{\hspace{0.17em}}x=1.\text{\hspace{0.17em}}$ y -intercept at $\text{\hspace{0.17em}}\left(0,6\right)$

Degree 5. Roots of multiplicity 2 at $\text{\hspace{0.17em}}x=3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ , and a root of multiplicity 1 at $\text{\hspace{0.17em}}x=–3.\text{\hspace{0.17em}}$ y -intercept at $\text{\hspace{0.17em}}\left(0,9\right)$

$f\left(x\right)=\frac{1}{3}{\left(x-3\right)}^{2}{\left(x-1\right)}^{2}\left(x+3\right)$

Degree 4. Root of multiplicity 2 at $\text{\hspace{0.17em}}x=4,\text{\hspace{0.17em}}$ and a roots of multiplicity 1 at $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=–2.\text{\hspace{0.17em}}$ y -intercept at $\text{\hspace{0.17em}}\left(0,\text{–}3\right).$

Degree 5. Double zero at $\text{\hspace{0.17em}}x=1,\text{\hspace{0.17em}}$ and triple zero at $\text{\hspace{0.17em}}x=3.\text{\hspace{0.17em}}$ Passes through the point $\text{\hspace{0.17em}}\left(2,15\right).$

$f\left(x\right)=-15{\left(x-1\right)}^{2}{\left(x-3\right)}^{3}$

Degree 3. Zeros at $\text{\hspace{0.17em}}x=4,$ $\text{\hspace{0.17em}}x=3,$ and $\text{\hspace{0.17em}}x=2.\text{\hspace{0.17em}}$ y -intercept at $\text{\hspace{0.17em}}\left(0,-24\right).$

Degree 3. Zeros at $\text{\hspace{0.17em}}x=-3,$ $\text{\hspace{0.17em}}x=-2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=1.\text{\hspace{0.17em}}$ y -intercept at $\text{\hspace{0.17em}}\left(0,12\right).$

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

Degree 5. Roots of multiplicity 2 at $\text{\hspace{0.17em}}x=-3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and a root of multiplicity 1 at $\text{\hspace{0.17em}}x=-2.$

y -intercept at

Degree 4. Roots of multiplicity 2 at $\text{\hspace{0.17em}}x=\frac{1}{2}\text{\hspace{0.17em}}$ and roots of multiplicity 1 at $\text{\hspace{0.17em}}x=6\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-2.$

y -intercept at $\text{\hspace{0.17em}}\left(0,18\right).$

$f\left(x\right)=-\frac{3}{2}{\left(2x-1\right)}^{2}\left(x-6\right)\left(x+2\right)$

Double zero at $\text{\hspace{0.17em}}x=-3\text{\hspace{0.17em}}$ and triple zero at $\text{\hspace{0.17em}}x=0.\text{\hspace{0.17em}}$ Passes through the point $\text{\hspace{0.17em}}\left(1,32\right).$

## Technology

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.

$f\left(x\right)={x}^{3}-x-1$

local max local min

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

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

global min

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

$f\left(x\right)={x}^{4}-{x}^{3}+1$

global min

## Extensions

For the following exercises, use the graphs to write a polynomial function of least degree.

$f\left(x\right)={\left(x-500\right)}^{2}\left(x+200\right)$

## Real-world applications

For the following exercises, write the polynomial function that models the given situation.

A rectangle has a length of 10 units and a width of 8 units. Squares of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of $\text{\hspace{0.17em}}x.$

$f\left(x\right)=4{x}^{3}-36{x}^{2}+80x$

Consider the same rectangle of the preceding problem. Squares of $\text{\hspace{0.17em}}2x\text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}2x\text{\hspace{0.17em}}$ units are cut out of each corner. Express the volume of the box as a polynomial in terms of $\text{\hspace{0.17em}}x.$

A square has sides of 12 units. Squares by units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a function in terms of $\text{\hspace{0.17em}}x.$

$f\left(x\right)=4{x}^{3}-36{x}^{2}+60x+100$

A cylinder has a radius of $\text{\hspace{0.17em}}x+2\text{\hspace{0.17em}}$ units and a height of 3 units greater. Express the volume of the cylinder as a polynomial function.

A right circular cone has a radius of $\text{\hspace{0.17em}}3x+6\text{\hspace{0.17em}}$ and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is $\text{\hspace{0.17em}}V=\frac{1}{3}\pi {r}^{2}h\text{\hspace{0.17em}}$ for radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ and height $\text{\hspace{0.17em}}h.$

$f\left(x\right)=\pi \left(9{x}^{3}+45{x}^{2}+72x+36\right)$

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?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
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×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali