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3.2 Graphs of Polynomial Functions of Higher Degree
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Continuous Polynomial Function- no breaks, holes, or gaps;
only smooth rounded turns (no sharp turns like )
xxf
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Sketching by hand…
*** Must be in Standard Form***
There will be 2 cases
01
1... axaxaxf nn
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Case 1:If n is even (therefore even degree), the graph has a shape similar to
The Right and Left Hand Behavior:
If the leading coefficient is positive , the graph rises to the left and right.
If the leading coefficient is negative , the graph falls to the left and right.
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Case 2:If n is odd (therefore odd degree), the graph has a shape similar to
The Right and Left Hand Behavior:
If the leading coefficient is positive , the graph falls to the left and rises to the right.
If the leading coefficient is negative , the graph rises to the left and falls to the right.
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Describe the right-hand and left-hand behavior.
xxxf
xxxf
xxxf
5
24
3
.3
45.2
4.1
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x - a is a factor of
If x = a is a zero, then:x = a is a solution for
Zeros of Polynomial Functions (x values):
For a polynomial of degree n, f has at most n -1 turning points
(where the graph goes from increasing to decreasing and vice versa a.k.a EXTREMA)
and f has at most n real zeros.
0xf xf
(a, 0) is an x intercept of the graph of f.
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Find all zeros. 24 22 xxxf
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Given a factor of , there is a repeated zero at x = a, of multiplicity k.
• If k is odd, the graph crosses the x axis at x = a • If k is even, the graph touches the x axis at x =
a (bounces)
1, kax k
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Steps:
• Determine the right and left hand behavior.• Factor to determine the zeros.• Use the multiplicity factor to determine how
the zeros affects the graph (crosses through or touches x axis).
• Sketch the graph.
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Graph Without a Calculator 34 43 xxxf
xxxxf2
962 23
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Intermediate Value Theorem
If a<b and , on [a,b], f takes on every value between . This can be used to approximate the real zero.
Example:
bfaf bfaf &
123 xxxf