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: Rice University

### Graphing Polynomial Functions with a Calculator

You have already 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.

Let's 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*}

Now, let's find the zeros, maximum, and minimum of the function from the previous problem above.

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 the previous problem above, 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).

Finally, let's 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 to find 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*} into your calculator, you find that the maximum occurs when \begin{align*}x = 6.15105\end{align*}. At that value of *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*}intercept: (0, -4)

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*}intercept: (0, -8)

minimum: (-1.11, 4.41)

maximum: (0.08, -8.12)

#### Example 4

Find the domain and range of Examples 2 and 3.

The domain of Example 2 is all real numbers and the range is all real numbers less than the maximum; \begin{align*}(-\infty, 113.12]\end{align*}. The domain and range of Example 3 are all real numbers.

### 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*}
- \begin{align*}h(x)=-\frac{1}{4}x^4-2x^3-\frac{13}{4} x^2-8x-9\end{align*}
- \begin{align*}y=x^3-8\end{align*}
- \begin{align*}g(x)=-x^3-11x^2-14x+10\end{align*}
- \begin{align*}f(x)=2x^4+3x^3-26x^2-3x+54\end{align*}
- \begin{align*}y=x^4+2x^3-5x^2-12x-6\end{align*}
- 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*}, 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*} degree polynomial has \begin{align*}n\end{align*} 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*}. 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.