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Slide 1:

Copyright © Cengage Learning. All rights reserved. 2 Polynomial and Rational Functions

Slide 2:

2.2 POLYNOMIAL FUNCTIONS OF HIGHER DEGREE Copyright © Cengage Learning. All rights reserved.

What You Should Learn:

Use transformations to sketch graphs of polynomial functions. Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions. Find and use zeros of polynomial functions as sketching aids. Use the Intermediate Value Theorem to help locate zeros of polynomial functions. What You Should Learn

Slide 4:

Graphs of Polynomial Functions

Graphs of Polynomial Functions:

Graphs of Polynomial Functions In this section, we will study basic features of the graphs of polynomial functions. The first feature is that the graph of a polynomial function is continuous. Essentially, this means that the graph of a polynomial function has no breaks, holes, or gaps, as shown in Figure 2.11(a). Figure 2.11(a) Polynomial functions have continuous graphs.

Graphs of Polynomial Functions:

Graphs of Polynomial Functions The graph shown in Figure 2.11(b) is an example of a piecewise defined function that is not continuous. Functions with graphs that are not continuous are not polynomial functions. Figure 2.11(b)

Graphs of Polynomial Functions:

Graphs of Polynomial Functions The second feature is that the graph of a polynomial function has only smooth, rounded turns, as shown in Figure 2.12. Polynomial functions have graphs with smooth, rounded turns. Figure 2.12

Graphs of Polynomial Functions:

Graphs of Polynomial Functions A polynomial function cannot have a sharp turn. For instance, the function given by f ( x ) = | x |, which has a sharp turn at the point (0, 0), as shown in Figure 2.13, is not a polynomial function. The graphs of polynomial functions of degree greater than 2 are more difficult to analyze than the graphs of polynomials of degree 0, 1, or 2. Graphs of polynomial functions cannot have sharp turns. Figure 2.13

Graphs of Polynomial Functions:

Graphs of Polynomial Functions However, using the features presented in this section, coupled with your knowledge of point plotting, intercepts, and symmetry, you should be able to make reasonably accurate sketches by hand . The polynomial functions that have the simplest graphs are monomials of the form f ( x ) = x n , where n is an integer greater than zero.

Graphs of Polynomial Functions:

Graphs of Polynomial Functions From Figure 2.14, you can see that when n is even , the graph is similar to the graph of f ( x ) = x 2 , and when n is odd , the graph is similar to the graph of f ( x ) = x 3 . If n is even, the graph of y = x n touches the axis at the x -intercept. If n is old, the graph of y = x n crosses the axis at the x -intercept. Figure 2.14 (a) (b)

Graphs of Polynomial Functions:

Graphs of Polynomial Functions Moreover, the greater the value of n , the flatter the graph near the origin. Polynomial functions of the form f ( x ) = x n are often referred to as power functions.

Example 1 – Sketching Transformations of Polynomial Functions:

Example 1 – Sketching Transformations of Polynomial Functions Sketch the graph of each function. a. f ( x ) = – x 5 b. h ( x ) = ( x + 1) 4 Solution: a. Because the degree of f ( x ) = – x 5 is odd, its graph is similar to the graph of y = x 3 . In Figure 2.15, note that the negative coefficient has the effect of reflecting the graph in the x -axis. Figure 2.15

Example 1 – Solution:

Example 1 – Solution b. The graph of h ( x ) = ( x + 1) 4 , as shown in Figure 2.16, is a left shift by one unit of the graph of y = x 4 . cont’d Figure 2.16

Slide 14:

The Leading Coefficient Test

The Leading Coefficient Test:

The Leading Coefficient Test In Example 1, note that both graphs eventually rise or fall without bound as x moves to the right. Whether the graph of a polynomial function eventually rises or falls can be determined by the function’s degree (even or odd) and by its leading coefficient, as indicated in the Leading Coefficient Test.

The Leading Coefficient Test:

The Leading Coefficient Test

The Leading Coefficient Test:

The Leading Coefficient Test cont’d

Example 2 – Applying the Leading Coefficient Test:

Example 2 – Applying the Leading Coefficient Test Describe the right-hand and left-hand behavior of the graph of each function. a. f ( x ) = – x 3 + 4 x b. f ( x ) = x 4 – 5 x 2 + 4 c. f ( x ) = x 5 – x

Example 2(a) – Solution:

Example 2(a) – Solution Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right, as shown in Figure 2.17. Figure 2.17

Example 2(b) – Solution:

Example 2(b) – Solution Because the degree is even and the leading coefficient is positive, the graph rises to the left and right, as shown in Figure 2.18. Figure 2.18 cont’d

Example 2(c) – Solution:

Example 2(c) – Solution Because the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right, as shown in Figure 2.19. Figure 2.19 cont’d

The Leading Coefficient Test:

The Leading Coefficient Test In Example 2, note that the Leading Coefficient Test tells you only whether the graph eventually rises or falls to the right or left. Other characteristics of the graph, such as intercepts and minimum and maximum points, must be determined by other tests.

Slide 23:

Zeros of Polynomial Functions

Zeros of Polynomial Functions:

Zeros of Polynomial Functions It can be shown that for a polynomial function f of degree n , the following statements are true. 1. The function f has, at most, n real zeros. 2. The graph of f has, at most, n – 1 turning points. (Turning points, also called relative minima or relative maxima, are points at which the graph changes from increasing to decreasing or vice versa.) Finding the zeros of polynomial functions is one of the most important problems in algebra.

Zeros of Polynomial Functions:

Zeros of Polynomial Functions There is a strong interplay between graphical and algebraic approaches to this problem. Sometimes you can use information about the graph of a function to help find its zeros, and in other cases you can use information about the zeros of a function to help sketch its graph. Finding zeros of polynomial functions is closely related to factoring and finding x -intercepts.

Zeros of Polynomial Functions:

Zeros of Polynomial Functions

Example 3 – Finding the Zeros of a Polynomial Function:

Example 3 – Finding the Zeros of a Polynomial Function Find all real zeros of f ( x ) = –2 x 4 + 2 x 2 . Then determine the number of turning points of the graph of the function. Solution: To find the real zeros of the function, set f ( x ) equal to zero and solve for x . –2 x 4 + 2 x 2 = 0 Set f ( x ) equal to 0.

Example 3 – Solution:

Example 3 – Solution –2 x 2 ( x 2 – 1) = 0 –2 x 2 ( x – 1)( x + 1) = 0 So, the real zeros are x = 0, x = 1, and x = –1. Because the function is a fourth-degree polynomial, the graph of f can have at most 4 – 1 = 3 turning points. Remove common monomial factor. Factor completely. cont’d

Zeros of Polynomial Functions:

Zeros of Polynomial Functions In Example 3, note that because the exponent is greater than 1, the factor –2 x 2 yields the repeated zero x = 0. Because the exponent is even, the graph touches the x -axis at x = 0, as shown in Figure 2.20. Figure 2.20

Zeros of Polynomial Functions:

Zeros of Polynomial Functions To graph polynomial functions, you can use the fact that a polynomial function can change signs only at its zeros. Between two consecutive zeros, a polynomial must be entirely positive or entirely negative.

Zeros of Polynomial Functions:

Zeros of Polynomial Functions This means that when the real zeros of a polynomial function are put in order, they divide the real number line into intervals in which the function has no sign changes. These resulting intervals are test intervals in which a representative x -value in the interval is chosen to determine if the value of the polynomial function is positive (the graph lies above the x -axis) or negative (the graph lies below the x -axis).

Slide 32:

The Intermediate Value Theorem

The Intermediate Value Theorem:

The Intermediate Value Theorem The next theorem, called the Intermediate Value Theorem, illustrates the existence of real zeros of polynomial functions. This theorem implies that if ( a , f ( a )) and ( b , f ( b )) are two points on the graph of a polynomial function such that f ( a )  f ( b ), then for any number d between f ( a ) and f ( b ) there must be a number c between a and b such that f ( c ) = d. (See Figure 2.25.) Figure 2.25

The Intermediate Value Theorem:

The Intermediate Value Theorem The Intermediate Value Theorem helps you locate the real zeros of a polynomial function in the following way. If you can find a value x = a at which a polynomial function is positive, and another value x = b at which it is negative, you can conclude that the function has at least one real zero between these two values.

The Intermediate Value Theorem:

The Intermediate Value Theorem For example, the function given by f ( x ) = x 3 + x 2 + 1 is negative when x = –2 and positive when x = –1. Therefore, it follows from the Intermediate Value Theorem that f must have a real zero somewhere between –2 and –1, as shown in Figure 2.26. Figure 2.26

The Intermediate Value Theorem:

The Intermediate Value Theorem By continuing this line of reasoning, you can approximate any real zeros of a polynomial function to any desired accuracy. This concept is further demonstrated in Example 6.

Example 6 – Approximating a Zero of a Polynomial Function:

Example 6 – Approximating a Zero of a Polynomial Function Use the Intermediate Value Theorem to approximate the real zero of f ( x ) = x 3 – x 2 + 1. Solution: Begin by computing a few function values, as follows.

Example 6 – Solution:

Example 6 – Solution Because f (–1) is negative and f (0) is positive, you can apply the Intermediate Value Theorem to conclude that the function has a zero between –1 and 0. To pinpoint this zero more closely, divide the interval [–1, 0] into tenths and evaluate the function at each point. When you do this, you will find that f (–0.8) = –0.152 and f (–0.7) = 0.167. cont’d

Example 6 – Solution:

Example 6 – Solution So, f must have a zero between – 0.8 and – 0.7, as shown in Figure 2.27. For a more accurate approximation, compute function values between f (–0.8) and f (–0.7) and apply the Intermediate Value Theorem again. By continuing this process, you can approximate this zero to any desired accuracy. Figure 2.27 cont’d

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