# 3.4 Graphs of polynomial functions  (Page 5/13)

 Page 5 / 13

Given a polynomial function, sketch the graph.

1. Find the intercepts.
2. Check for symmetry. If the function is an even function, its graph is symmetrical about the $\text{\hspace{0.17em}}y\text{-}$ axis, that is, $\text{\hspace{0.17em}}f\left(-x\right)=f\left(x\right).\text{\hspace{0.17em}}$ If a function is an odd function, its graph is symmetrical about the origin, that is, $\text{\hspace{0.17em}}f\left(-x\right)=-f\left(x\right).$
3. Use the multiplicities of the zeros to determine the behavior of the polynomial at the $\text{\hspace{0.17em}}x\text{-}$ intercepts.
4. Determine the end behavior by examining the leading term.
5. Use the end behavior and the behavior at the intercepts to sketch a graph.
6. Ensure that the number of turning points does not exceed one less than the degree of the polynomial.
7. Optionally, use technology to check the graph.

## Sketching the graph of a polynomial function

Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)=-2{\left(x+3\right)}^{2}\left(x-5\right).$

This graph has two $\text{\hspace{0.17em}}x\text{-}$ intercepts. At $\text{\hspace{0.17em}}x=-3,\text{\hspace{0.17em}}$ the factor is squared, indicating a multiplicity of 2. The graph will bounce at this $\text{\hspace{0.17em}}x\text{-}$ intercept. At $\text{\hspace{0.17em}}x=5,\text{\hspace{0.17em}}$ the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.

The y -intercept is found by evaluating $\text{\hspace{0.17em}}f\left(0\right).$

The $\text{\hspace{0.17em}}y\text{-}$ intercept is $\text{\hspace{0.17em}}\left(0,90\right).$

Additionally, we can see the leading term, if this polynomial were multiplied out, would be $\text{\hspace{0.17em}}-2{x}^{3},\text{\hspace{0.17em}}$ so the end behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs increasing as the inputs approach negative infinity. See [link] .

To sketch this, we consider that:

• As $\text{\hspace{0.17em}}x\to -\infty \text{\hspace{0.17em}}$ the function $\text{\hspace{0.17em}}f\left(x\right)\to \infty ,\text{\hspace{0.17em}}$ so we know the graph starts in the second quadrant and is decreasing toward the $\text{\hspace{0.17em}}x\text{-}$ axis.
• Since $\text{\hspace{0.17em}}f\left(-x\right)=-2{\left(-x+3\right)}^{2}\left(-x–5\right)\text{\hspace{0.17em}}$ is not equal to $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ the graph does not display symmetry.
• At $\text{\hspace{0.17em}}\left(-3,0\right),\text{\hspace{0.17em}}$ the graph bounces off of the $\text{\hspace{0.17em}}x\text{-}$ axis, so the function must start increasing.

At $\text{\hspace{0.17em}}\left(0,90\right),\text{\hspace{0.17em}}$ the graph crosses the $\text{\hspace{0.17em}}y\text{-}$ axis at the $\text{\hspace{0.17em}}y\text{-}$ intercept. See [link] .

Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at $\text{\hspace{0.17em}}\left(5,0\right).\text{\hspace{0.17em}}$ See [link] .

As $\text{\hspace{0.17em}}x\to \infty \text{\hspace{0.17em}}$ the function $\text{\hspace{0.17em}}f\left(x\right)\to \mathrm{-\infty },\text{\hspace{0.17em}}$ so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.

Using technology, we can create the graph for the polynomial function, shown in [link] , and verify that the resulting graph looks like our sketch in [link] .

Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{4}x{\left(x-1\right)}^{4}{\left(x+3\right)}^{3}.$

## Using the intermediate value theorem

In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the x -axis, we can confirm that there is a zero between them. Consider a polynomial function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ whose graph is smooth and continuous. The Intermediate Value Theorem states that for two numbers $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ in the domain of $\text{\hspace{0.17em}}f,$ if $\text{\hspace{0.17em}}a and $f\left(a\right)\ne f\left(b\right),$ then the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ takes on every value between $\text{\hspace{0.17em}}f\left(a\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(b\right).\text{\hspace{0.17em}}$ We can apply this theorem to a special case that is useful in graphing polynomial functions. If a point on the graph of a continuous function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}x=a\text{\hspace{0.17em}}$ lies above the $\text{\hspace{0.17em}}x\text{-}$ axis and another point at $\text{\hspace{0.17em}}x=b\text{\hspace{0.17em}}$ lies below the $\text{\hspace{0.17em}}x\text{-}$ axis, there must exist a third point between $\text{\hspace{0.17em}}x=a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=b\text{\hspace{0.17em}}$ where the graph crosses the $\text{\hspace{0.17em}}x\text{-}$ axis. Call this point This means that we are assured there is a solution $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ where $f\left(c\right)=0.$

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