# 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:

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

### Watch This

James Sousa: Ex: Solve a Polynomial Equation Using a Graphing Calculator (Approximate Solutions)

### Guidance

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.

#### Example A

Graph

**Solution:** *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

#### Example B

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

**Solution:** 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 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).

#### Example C

Find the

**Solution:** 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 “

**Intro Problem Revisit** 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).

### Guided Practice

Graph and find the critical values of the following functions.

1.

2.

3. Find the domain and range of the previous two functions.

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

#### Answers

Use the steps given in Examples

1. 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)

2. zeros: -1.413, -0.682, 0.672

minimum: (-1.11, 4.41)

maximum: (0.08, -8.12)

3. The domain of #1 is all real numbers and the range is all real numbers less than the maximum;

4. There are three irrational solutions and two imaginary solutions.

### Explore More

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 Explore More Problems

To view the Explore More answers, open this PDF file and look for section 6.14.

### Image Attributions

## Description

## Learning Objectives

Here you'll learn how to graph polynomial functions and find critical values using a graphing calculator.

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## Date Created:

Mar 12, 2013## Last Modified:

Jun 04, 2015## Vocabulary

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