# 6.14: Graphing Polynomial Functions with a Graphing Calculator

**At Grade**Created by: CK-12

**Practice**Graphing Calculator to Analyze Polynomial Functions

To make a fair race between a dragster and a funny car, a scientist devised the following polynomial equation:

\begin{align*}f(x) = 71.682x -60.427x^2 + 84.710x^3 -27.769x^4 + 4.296x^5 - 0.262x^6\end{align*}

Source: http://ceee.rice.edu/Books/CS/chapter3/data1.html

### Graphing Polynomial Functions with a Calculator

In the Quadratic Functions chapter, you used the graphing calculator to graph parabolas. Now, we will expand upon that knowledge and graph higher-degree polynomials. Then, we will use the graphing calculator to find the zeros, maximums and minimums.

#### Use your graphing calculator to solve the following problems

Graph \begin{align*}f(x)=x^3+x^2-8x-8\end{align*}

*These instructions are for a TI-83 or 84*. First, press \begin{align*}Y=\end{align*}**ENTER.** Now, in \begin{align*}Y1\end{align*}**GRAPH**.

To adjust the window, press **ZOOM**. To get the typical -10 to 10 screen (for both axes), press **6:ZStandard.** To zoom out, press **ZOOM, 3:ZoomOut, ENTER, ENTER.** For this particular function, the window needs to go from -15 to 15 for both \begin{align*}x\end{align*}

Find the zeros, maximum, and minimum of the function from Example A.

To find the zeros, press \begin{align*}2^{nd}\end{align*}**TRACE** to get the **CALC** menu. Select **2:Zero** and you will be asked “Left Bound?” by the calculator. Move the cursor (by pressing the \begin{align*}\uparrow\end{align*}**ENTER.** Then, it will ask “Right Bound?” Move the cursor just to the right of that zero. Press **ENTER.** The calculator will then ask “Guess?” At this point, you can enter in what you think the zero is and press **ENTER** again. Then the calculator will give you the exact zero. For the graph from Example A, you will need to repeat this three times. The zeros are -2.83, -1, and 2.83.

To find the minimum and maximum, the process is almost identical to finding zeros. Instead of selecting **2:Zero**, select **3:min** or **4:max**. The minimum is (1.33, -14.52) and the maximum is (-2, 4).

Find the \begin{align*}y-\end{align*}

If you decide not to use the calculator, plug in zero for \begin{align*}x\end{align*}

\begin{align*}f(0) &= 0^3+0^2 - 8 \cdot 0 - 8\\
&= -8\end{align*}

Using the graphing calculator, press \begin{align*}2^{nd}\end{align*}**TRACE** to get the **CALC** menu. Select **1:value.** \begin{align*}X=\end{align*}**CLEAR** to remove it. Then press **0** and **ENTER.** The calculator should then say “\begin{align*}Y=-8\end{align*}

### Examples

#### Example 1

Earlier, you were asked what is the maximum point of the function's graph.

If you plug the equation \begin{align*}f(x) = 71.682x -60.427x^2 + 84.710x^3 -27.769x^4 + 4.296x^5 - 0.262x^6\end{align*}*x*, *f(x)* equals 1754.43. Therefore the maximum point of the function's graph is (6.15105, 1754.43).

Graph and find the critical values of the following functions.

#### Example 2

\begin{align*}f(x)=-\frac{1}{3}x^4-x^3+10x^2+25x-4\end{align*}

zeros: -5.874, -2.56, 0.151, 5.283

\begin{align*}y-\end{align*}

minimum: (-1.15, -18.59)

local maximum: (-4.62, 40.69)

absolute maximum: (3.52, 113.12)

#### Example 3

\begin{align*}g(x)=2x^5-x^4+6x^3+18x^2-3x-8\end{align*}

zeros: -1.413, -0.682, 0.672

\begin{align*}y-\end{align*}

minimum: (-1.11, 4.41)

maximum: (0.08, -8.12)

#### Example 4

Find the domain and range of the previous two functions.

The domain of #1 is all real numbers and the range is all real numbers less than the maximum; \begin{align*}(-\infty, 113.12]\end{align*}

#### Example 5

Describe the types of solutions, as specifically as possible, for question 2.

There are three irrational solutions and two imaginary solutions.

### Review

Graph questions 1-6 on your graphing calculator. Sketch the graph in an appropriate window. Then, find all the critical values, domain, range, and describe the end behavior.

- \begin{align*}f(x)=2x^3+5x^2-4x-12\end{align*}
f(x)=2x3+5x2−4x−12 - \begin{align*}h(x)=-\frac{1}{4}x^4-2x^3-\frac{13}{4} x^2-8x-9\end{align*}
h(x)=−14x4−2x3−134x2−8x−9 - \begin{align*}y=x^3-8\end{align*}
y=x3−8 - \begin{align*}g(x)=-x^3-11x^2-14x+10\end{align*}
g(x)=−x3−11x2−14x+10 - \begin{align*}f(x)=2x^4+3x^3-26x^2-3x+54\end{align*}
f(x)=2x4+3x3−26x2−3x+54 - \begin{align*}y=x^4+2x^3-5x^2-12x-6\end{align*}
y=x4+2x3−5x2−12x−6 - What are the types of solutions in #2?
- Find the two imaginary solutions in #3.
- Find the exact values of the irrational roots in #5.

Determine if the following statements are SOMETIMES, ALWAYS, or NEVER true. Explain your reasoning.

- The range of an even function is \begin{align*}(-\infty, max]\end{align*}
(−∞,max] , where*max*is the maximum of the function. - The domain and range of all odd functions are all real numbers.
- A function can have exactly three imaginary solutions.
- An \begin{align*}n^{th}\end{align*}
nth degree polynomial has \begin{align*}n\end{align*}n real solutions. - The parent graph of any polynomial function has one zero.
**Challenge**The exact value for one of the zeros in #2 is \begin{align*}-4+\sqrt{7}\end{align*}−4+7√ . What is the exact value of the other root? Then, use this information to find the imaginary roots.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.14.

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