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

 Page 9 / 13

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

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

$f\left(x\right)={x}^{3}-9x,\text{\hspace{0.17em}}$ between $\text{\hspace{0.17em}}x=-4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-2.$

$f\left(x\right)={x}^{3}-9x,\text{\hspace{0.17em}}$ between $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=4.$

$f\left(2\right)=–10\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(4\right)=28.$ Sign change confirms.

$f\left(x\right)={x}^{5}-2x,\text{\hspace{0.17em}}$ 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, $\left(–4, 0\right)$ with multiplicity 1, $y\text{-}$ intercept $\left(0, 4\right)$ . As $x\to -\infty$ , $f\left(x\right)\to -\infty$ , as $x\to \infty$ , $f\left(x\right)\to \infty$ .

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

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

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

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

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

x -intercepts $\left(0, 0\right),\phantom{\rule{0.2em}{0ex}}\left(–2, 0\right),\phantom{\rule{0.2em}{0ex}}\left(4, 0\right)$ with multiplicity 1, $y$ -intercept $\left(0, 0\right).$ As $x\to -\infty$ , $f\left(x\right)\to \infty$ , as $x\to \infty$ , $f\left(x\right)\to -\infty .$

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 $\text{\hspace{0.17em}}h.$

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

The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations