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

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

These instructions are for a TI-83 or 84. First, press **ENTER.** Now, in **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

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

To find the zeros, press **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 **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

If you decide not to use the calculator, plug in zero for

Using the graphing calculator, press **TRACE** to get the **CALC** menu. Select **1:value.** **CLEAR** to remove it. Then press **0** and **ENTER.** The calculator should then say “

### Examples

#### Example 1

Earlier, you were asked to find the maximum point of the function's graph.

If you plug the equation *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

zeros: -5.874, -2.56, 0.151, 5.283

minimum: (-1.15, -18.59)

local maximum: (-4.62, 40.69)

absolute maximum: (3.52, 113.12)

#### Example 3

zeros: -1.413, -0.682, 0.672

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;

### 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.

f(x)=2x3+5x2−4x−12 h(x)=−14x4−2x3−134x2−8x−9 y=x3−8 g(x)=−x3−11x2−14x+10 f(x)=2x4+3x3−26x2−3x+54 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
(−∞,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
nth degree polynomial hasn real solutions. - The parent graph of any polynomial function has one zero.
**Challenge**The exact value for one of the zeros in #2 is−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.