Transcript Ch 3 Polynomial Functions

Copyright © 2011 Pearson Education, Inc.
Slide 3.5-1
Chapter 3: Polynomial Functions
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Complex Numbers
Quadratic Functions and Graphs
Quadratic Equations and Inequalities
Further Applications of Quadratic Functions and Models
Higher-Degree Polynomial Functions and Graphs
Topics in the Theory of Polynomial Functions (I)
Topics in the Theory of Polynomial Functions (II)
Polynomial Equations and Inequalities; Further
Applications and Models
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3.5 Higher Degree Polynomial Functions
and Graphs
Polynomial Function
A polynomial function of degree n in the variable x is
a function defined by
P( x)  an xn  an1xn1    a1x  a0
where each ai is real, an  0, and n is a whole number.
•
•
•
•
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
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3.5 Cubic Functions: Odd Degree
Polynomials
• The cubic function is a third degree polynomial of the
form
P( x)  ax3  bx 2  cx  d , a  0.
• In general, the graph of a cubic function will resemble one
of the following shapes.
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3.5 Quartic Functions: Even Degree
Polynomials
•
The quartic function is a fourth degree polynomial of
the form
P( x)  ax4  bx3  cx2  dx  e, a  0.
•
In general, the graph of a quartic function will resemble
one of the following shapes. The dashed portions
indicate irregular, but smooth, behavior.
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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
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3.5 Extrema
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3.5 Absolute and Local Extrema
Absolute and Local Extrema
Let c be in the domain of P. Then
(a) P(c) is an absolute maximum if P(c) > P(x) for all x in
the domain of P.
(b) P(c) is an absolute minimum if P(c) < P(x) for all x in
the domain of P.
(c) P(c) is a local maximum if P(c) > P(x) when x is near c.
(d) P(c) is a local minimum if P(c) < P(x) when x is near c.
The expression “near c” means that there is an open interval
in the domain of P conaining c, where P(c) satifies the
inequality.
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3.5 Identifying Local and Absolute Extrema
Example Consider the following graph.
(a) Identify and classify the local extreme points of f.
Local Min points: (a,b),(e,h); Local Max point: (c,d)
(b) Identify and classify the local extreme points of g.
Local Min point: (j,k); Local Max point: (m,n)
(c) Describe the absolute extreme points for f and g.
f has an absolute minimum value of h, but no
absolute maximum. g has no absolute extrema.
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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.
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3.5 End Behavior
Let axn be the dominating term of a polynomial function P.
n odd
n even
1.
1.
If a > 0, the graph of P falls on
the left and rises on the right.
x  , P( x)   and
x  , P ( x )  
x  , P ( x)  
2.
If a < 0, the graph of P rises on
the left and falls on the right.
x  , P( x)   and
If a > 0, the graph of P opens
up.
2.
If a < 0, the graph of P opens
down.
x  , P ( x )  
x  , P( x)  
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3.5 Determining End Behavior
Match each function with its graph.
f ( x)  x 4  x 2  5 x  4
g ( x)   x 6  x 2  3x  4
h( x)  3x 3  x 2  2 x  4 and k ( x)  7 x 7  x  4
A.
C.
B.
D.
Solution: f matches C, g matches A, h matches B, k matches D.
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3.5 x-Intercepts (Real Zeros)
Number of x-Intercepts (Real Zeros) of a Polynomial Function
The graph of a polynomial function of degree n will have at most n
x-intercepts (real zeros).
Example Find the x-intercepts of P( x)  x 3  5 x 2  5 x  2.
Solution By using the
graphing calculator in a
standard viewing window,
the x-intercepts (real zeros)
are –2, approximately
–3.30, and approximately 0.30.
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3.5 Analyzing a Polynomial Function
P( x)  x 5  2 x 4  x 3  x 2  x  4
(a)
(b)
(c)
(d)
Determine its domain.
Determine its range.
Use its graph to find approximations of local extrema.
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 (–, ).
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3.5 Analyzing a Polynomial Function
(c) Two extreme points that we approximate using a graphing
calculator: local maximum point (– 2.02,10.01), and
local minimum point (0.41, – 4.24).
Looking Ahead to Calculus
The 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.
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3.5 Analyzing a Polynomial Function
(d) We use calculator methods to find that the x-intercepts are
–1 (exact), 1.14(approximate), and –2.52 (approximate).
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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 reveal the correct end
behavior.
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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?
P( x)  x  36 x  288x  256
6
4
2
Solution
Since P is a sixth degree
polynomial, it can have
up to 6 x-intercepts. Try
a window of [-8,8] by
[-1000,600].
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