# 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.

how fast can i understand functions without much difficulty
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