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Lesson

Throughout this chapter, you have been developing skills to help you graph various polynomial functions. Most of your sketching, however, has been limited to quadratic, cubic and quartic functions. Just as you may expect, the skills used here can also be used to sketch the graph of any function. Although the shapes may vary, there are many common characteristics.

The steps used in graphing functions are summarized for those of you who wish to review the process.

Before beginning the Investigation, use your graphing skills to sketch the curve given by f(x) = x⁴ + 6x³ + 7x² - 6x - 8. Once the task is completed, you may check your solution.

Read the introduction to Investigation 11 on page 105 of your text.

Notebook Entry:Record the definition of a critical point. 

Think back to the previous section for a moment. Visualize the graph of a slope function; any one will suffice. 

What do the x-intercepts of the slope function tell you about the graph of the original polynomial function? 

Examining critical points, you may have surmised, is an important part of curve sketching. You have already seen that local maxima and minima often occur at points where the derivative is zero. How, you may wonder, can the derivative of a function ever be undefined at a point? You must remember, however, that the derivative of a function at a point is defined as a limit. If this limit does not exist, then the derivative is undefined at the point. 

Consider, for example, the function f(x) = x² where the domain is restricted to the closed interval [0, 1]. The highest possible point, global maximum, on this graph is (1, 1). Now try to find the derivative at this point. 

Verify your solution.

You should now be ready to begin Investigation 11. Don't forget to answer the Investigation Questions as well. A few hints provided for some of the steps are to be used only if absolutely necessary.

• Step A
• Step B
• Step C
• Step D

Activity

C.Y.U. pages 106 - 108 #'s 6 - 14

When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.

Test Yourself

Find the critical points of the following functions.

1. y = 2x² + 2x - 1
2. y = 3x³ - 4x

Solutions

1. y ^/ = 4x + 2
      0 = 4x + 2
     x = -           critical point:

2. y ^/ = 9x² - 4
      0 = 9x² - 4
      x =            critical points: