# 3.4 Graphs of polynomial functions

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In this section, you will:
• Recognize characteristics of graphs of polynomial functions.
• Use factoring to ﬁnd zeros of polynomial functions.
• Identify zeros and their multiplicities.
• Determine end behavior.
• Understand the relationship between degree and turning points.
• Graph polynomial functions.
• Use the Intermediate Value Theorem.

The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in [link] .

 Year 2006 2007 2008 2009 2010 2011 2012 2013 Revenues 52.4 52.8 51.2 49.5 48.6 48.6 48.7 47.1

The revenue can be modeled by the polynomial function

$R\left(t\right)=-0.037{t}^{4}+1.414{t}^{3}-19.777{t}^{2}+118.696t-205.332$

where $\text{\hspace{0.17em}}R\text{\hspace{0.17em}}$ represents the revenue in millions of dollars and $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ represents the year, with $\text{\hspace{0.17em}}t=6\text{\hspace{0.17em}}$ corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.

## Recognizing characteristics of graphs of polynomial functions

Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. [link] shows a graph that represents a polynomial function    and a graph that represents a function that is not a polynomial.

## Recognizing polynomial functions

Which of the graphs in [link] represents a polynomial function?

The graphs of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ are graphs of polynomial functions. They are smooth and continuous .

The graphs of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ are graphs of functions that are not polynomials. The graph of function $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ has a sharp corner. The graph of function $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is not continuous.

Do all polynomial functions have as their domain all real numbers?

Yes. Any real number is a valid input for a polynomial function.

## Using factoring to find zeros of polynomial functions

Recall that if $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is a polynomial function, the values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ for which $\text{\hspace{0.17em}}f\left(x\right)=0\text{\hspace{0.17em}}$ are called zeros    of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros .

We can use this method to find $\text{\hspace{0.17em}}x\text{-}$ intercepts because at the $\text{\hspace{0.17em}}x\text{-}$ intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases in this section:

1. The polynomial can be factored using known methods: greatest common factor and trinomial factoring.
2. The polynomial is given in factored form.
3. Technology is used to determine the intercepts.

Given a polynomial function $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}$ find the x -intercepts by factoring.

1. Set $\text{\hspace{0.17em}}f\left(x\right)=0.\text{\hspace{0.17em}}$
2. If the polynomial function is not given in factored form:
1. Factor out any common monomial factors.
2. Factor any factorable binomials or trinomials.
3. Set each factor equal to zero and solve to find the $\text{\hspace{0.17em}}x\text{-}$ intercepts.

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
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