Copyright © 2007 Pearson Education, Inc. Slide 3-1

Copyright © 2007 Pearson Education, Inc. Slide 3-1


Chapter 3: Polynomial Functions

3.1 Complex Numbers

3.2 Quadratic Functions and Graphs

3.3 Quadratic Equations and Inequalities

3.4 Further Applications of Quadratic Functions and Models

3.5 Higher Degree Polynomial Functions and Graphs

3.6 Topics in the Theory of Polynomial Functions (I)

3.7 Topics in the Theory of Polynomial Functions (II)

3.8 Polynomial Equations and Inequalities; Further Applications and Models


3.5 Higher Degree Polynomial Functions and Graphs

• an is called the leading coefficient

• anxn is called the dominating term

• a0 is called the constant term

• P(0) = a0 is the y-intercept of the graph of P

Polynomial Function

A polynomial function of degree n in the variable x is a function defined by

where each ai is real, an 0, and n is a whole number.01

11)( axaxaxaxP n

nn

n


3.5 Cubic Functions: Odd Degree Polynomials

• The cubic function is a third degree polynomial of the form

• In general, the graph of a cubic function will resemble one of the following shapes.

.0 ,)( 23 adcxbxaxxP


3.5 Quartic Functions: Even Degree Polynomials

• The quartic function is a fourth degree polynomial of the form

• In general, the graph of a quartic function will resemble one of the following shapes. The dashed portions indicate irregular, but smooth, behavior.

.0,)( 234 aedxcxbxaxxP


3.5 Extrema

• Turning points – where the graph of a function changes from increasing to decreasing or vice versa

• Local maximum point – highest point or “peak” in an interval

– function values at these points are called local maxima

• Local minimum point – lowest point or “valley” in an interval

– function values at these points are called local minima

• Extrema – plural of extremum, includes all local maxima and local minima


3.5 Extrema


3.5 Identifying Local and Absolute Extrema

Example Consider the following graph.

(a) Name and classify the local extrema of f.

(b) Name and classify the local extrema of g.

(c) Describe the absolute extrema for f and g.

Local Min points: (a,b),(e,h); Local Max point: (c,d)

Local Min point: (j,k); Local Max point: (m,n)

f has an absolute minimum value of h, but no absolute maximum. g has no absolute extrema.


3.5 Number of Local Extrema

• A linear function has degree 1 and no local extrema.

• A quadratic function has degree 2 with one extreme point.

• A cubic function has degree 3 with at most two local extrema.

• A quartic function has degree 4 with at most three local extrema.

• Extending this idea:

Number of Turning Points

The number of turning points of the graph of a polynomial function of degree n 1 is at most n – 1.


Let axn be the dominating term of a polynomial function P.

3.5 End Behavior

n odd1. If a > 0, the graph of P falls on

the left and rises on the right.

2. If a < 0, the graph of P rises on the left and falls on the right.

n even1. If a > 0, the graph of P opens

up.

2. If a < 0, the graph of P opens down.

)(,

and )(,

xPx

xPx

)(,

and )(,

xPx

xPx

)(, xPx

)(, xPx


3.5 Determining End Behavior

Match each function with its graph.

Solution:

and423)(45)(

23

24

xxxxh

xxxxf

47)(43)(

7

26

xxxkxxxxg

f matches C, g matches A, h matches B, k matches D.

A. B.

C. D.


3.5 x-Intercepts (Real Zeros)

Example Find the x-intercepts of

Solution By using the graphing calculator in a standard viewing window, the x-intercepts (real zeros) are –2, approximately –3.30, and approximately .30.

Number Of x-Intercepts of a Polynomial Function

A polynomial function of degree n will have a maximum of n x- intercepts (real zeros).

.255)( 23 xxxxP


3.5 Analyzing a Polynomial Function

(a) Determine its domain.

(b) Determine its range.

(c) Use its graph to find approximations of local extrema.

(d) Use its graph to find the approximate and/or exact

x- intercepts.

Solution

(a) Since P is a polynomial, its domain is (–, ).

(b) Because it is of odd degree, its range is (–, ).

42)( 2345 xxxxxxP



(c) Two extreme points that we approximate using a graphing calculator: local maximum point (– 2.02,10.01), andlocal minimum point (.41, – 4.24).

Looking Ahead to CalculusThe derivative gives the slope of f at any value in the domain. The slope at local extrema is 0 since the tangent line is horizontal.



(d) We use calculator methods to find that the x-intercepts are –1 (exact), 1.14(approximate), and –2.52 (approximate).


3.5 Comprehensive Graphs

• The most important features of the graph of a polynomial function are:

1. intercepts,

2. extrema,

3. end behavior.

• A comprehensive graph of a polynomial function will exhibit the following features:

1. all x-intercepts (if any),

2. the y-intercept,

3. all extreme points (if any),

4. enough of the graph to exhibit end behavior.


3.5 Determining the Appropriate Graphing Window

The window [–1.25,1.25] by [–400,50] is used in the following graph. Is this a comprehensive graph?

SolutionSince P is a sixth degree polynomial, it can have up to 6 x-intercepts. Trya window of [-8,8] by[-1000,600].

25628836)( 246 xxxxP

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Copyright © 2007 Pearson Education, Inc. Slide 3-1