Polynomial Functions
and their Graphs
Section 3.1
General Shape of Polynomial Graphs
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The graph of polynomials are smooth,
unbroken lines or curves, with no sharp
corners or cusps (see p. 251).
Every Polynomial function is defined and
continuous for all real numbers.
Review

General polynomial formula
P( x)  an x  an1x
n
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

n1
 ...  a1x  a0
a0, a1, … ,an are constant coefficients
n is the degree of the polynomial
Standard form is for descending powers of x
anxn is said to be the “leading term”
Family of Polynomials

Constant polynomial functions
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Linear polynomial functions
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f(x) = a
f(x) = mx + b
Quadratic polynomial functions

f(x) = ax2 + bx + c
Family of Polynomials
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Cubic polynomial functions
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f(x) = a x3 + b x2 + c x + d
3rd degree polynomial
Quartic polynomial functions
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f(x) = a x4 + b x3 + c x2+ d x + e
4th degree polynomial
Polynomial “End Behavior”

Consider what happens when x gets very
large in positive and negative direction
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Called “end behavior”
Also “long-run” behavior
Basically, the leading term anxn
dominates the shape of graph
There are 4 possible scenarios:
End Behavior
Discuss end behavior for the following graphs:
4
3
P
(
x
)


2
x

5
x
 4x  7


P( x)  2x5  x4  6x2 10

P( x)  3x  5x  2 x
5
3
Compare Graph Behavior
Consider the following graphs:
 f(x) = x4 - 4x3 + 16x - 16
 g(x) = x4 - 4x3 - 4x2 +16x
 h(x) = x4 + x3 - 8x2 - 12x


Graph these on the window
-8 < x < 8
and
0 < y < 4000
Decide how these functions are alike or
different, based on the view of this graph
Compare Graph Behavior

From this view, they appear very similar
Compare “Short Run” Behavior

Now Change the window to be
-5 < x < 5 and -35 < y < 15

How do the functions appear to be different
from this view?
Compare Short Run Behavior
Differences?
 Real zeros
 Local extrema
 Complex zeros

Note: The standard form of the polynomials does
not give any clues as to this short run behavior of
the polynomials:
Using Zeros to Graph Polynomials

Consider the following polynomial:

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What will the zeros be for this polynomial?
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p(x) = (x - 2)(2x + 3)(x + 5)
x=2
x = -3/2
x = -5
How do you know?

Zero-Factor Property: If a*b = 0 then, we know
that either a = 0 or b = 0 (or both)
Guidelines to Graphing
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Zeros
Test Points (like table of signs)
End Behavior
Graph (a smooth curve through all
known points)
Intermediate Value Theorem

If P is a polynomial function and P(a) and
P(b) have opposite signs, then there is at
least one value c between a and b for which
P(c) = 0.
Theorem

Local Extrema of Polynomial Functions:

A polynomial function of degree n has at most
n - 1 local extrema.
Local Extrema (turning points)

Local Extrema – a point (x,y) on the graph
where the graph changes from increasing to
decreasing or vice-versa.
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