Transcript Graphs of Polynomial Functions

Graphs of Polynomial
Functions
The Polynomial Functions
• The key features of a polynomial graph
• Leading Coefficient Test to determine the
end behavior of graphs of polynomial
functions.
• Finding zeros of polynomial functions
• Determine a polynomial equation given
the zeros of the function.
Graphs of polynomial functions are continuous. That is, they
have no breaks, holes, or gaps.
f (x) = x3 – 5x2 + 4x + 4
y
x
y
y
x
x
continuous
not continuous
continuous
smooth
not smooth
polynomial
not polynomial
not polynomial
Polynomial functions are also smooth with rounded turns. Graphs
with points or cusps are not graphs of polynomial functions.
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A polynomial function is a function of the form
f ( x )  an x n  an 1 x n 1 
 a1 x  a0
where n is a nonnegative integer and a1, a2, a3, … an are
real numbers.
The polynomial function has a leading
coefficient an and degree n.
Examples: Find the leading coefficient and degree of each
polynomial function.
Polynomial Function
Leading Coefficient
Degree
f ( x )  2 x 5  3 x 3  5 x  1
-2
5
f ( x)  x3  6 x 2  x  7
1
3
f ( x )  14
14
0
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Classification of a Polynomial
Degree
Name
Example
n=0
constant
Y=3
n=1
linear
Y = 5x + 4
n=2
quadratic
Y = 2x2 + 3x - 2
n=3
cubic
Y = 5x3 + 3x2 – x + 9
n=4
quartic
Y = 3x4 – 2x3 + 8x2 – 6x + 5
n=5
quintic
Y = -2x5+3x4–x3+3x2–2x+6
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Graphs of Polynomial Functions
The polynomial functions that have the simplest
graphs are monomials of the form f(x) = xn,
where n is an integer greater than zero.
Polynomial functions of the form f (x) = x n, n  1 are
called power functions.
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y f (x) = x
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f
(x)
=
x
y
f (x) = x3
f (x) = x2
x
x
If n is even, their graphs
resemble the graph of
f (x) = x2.
If n is odd, their graphs
resemble the graph of
f (x) = x3.
Moreover, the greater the value of n, the
flatter the graph near the origin
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The Leading Coefficient Test
Polynomial functions have a domain of all real numbers.
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.
Leading Coefficient Test
As x grows positively or negatively without bound, the value
f (x) of the polynomial function
f (x) = anxn + an – 1xn – 1 + … + a1x + a0 (an  0)
grows positively or negatively without bound depending upon
the sign of the leading coefficient an and whether the degree n
is odd or even.
y
y
an positive
x
x
n odd
an negative
n even
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Using our calculator examine the
behavior of the polynomials
1.
y  x  3x  2
2.
y  3 x  2 x
2
3
3. f ( x )   x  4 x  1
4
3
4. y  3 x  2 x  1
5
2
5. f ( x )  3 x  2 x  8
6
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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.
Given the polynomials below, answer the following
A.
B.
C.
D.
E.
F.
What is the degree?
What is its leading coefficient?
How many “turns”(relative maximums or minimums) could it have (maximum)?
How many real zeros could it have (maximum)?
How would you describe the left and right behavior of the graph of the equation?
What are its intercepts (y for all, x for 1 & 2 only)?
Equations:
1. y   x 3  3x 2  2 x
2. f ( x)  x 4  2 x 2  8
3. y  3 x 5  2 x 2  1
4. f ( x)  4 x 6  4 x  1
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Repeated Zeros
If k is the largest integer for which (x – a) k is a factor of f (x)
and k > 1, then a is a repeated zero of multiplicity k.
1. If k is odd the graph of f (x) crosses the x-axis at (a, 0).
2. If k is even the graph of f (x) touches, but does not cross
through, the x-axis at (a, 0).
Example: Determine the multiplicity of the zeros
of f (x) = (x – 2)3(x +1)4.
y
Zero Multiplicity Behavior
crosses x-axis
3 odd
2
at (2, 0)
touches x-axis
–1
4 even
at (–1, 0)
x
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Example - Finding the Zeros of a Polynomial Function
Find all real zeros of
f(x) = –2x4 + 2x2.
Then determine the number of turning points of
the graph of the function.
Another example: Find all the real zeros and turning points of the
graph of f (x) = x 4 – x3 – 2x2.
Factor completely: f (x) = x 4 – x3 – 2x2 = x2(x + 1)(x – 2).
The real zeros are x = –1, x = 0, and x = 2.
These correspond to the
x-intercepts (–1, 0), (0, 0) and (2, 0).
The graph shows that
there are three turning points.
Since the degree is four, this is
Turning point
the maximum number possible.
y
Turning
point
x
Turning point
f (x) = x4 – x3 – 2x2
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Example continued: Sketch the graph of f (x) = 4x2 – x4.
3. Since f (–x) = 4(–x)2 – (–x)4 = 4x2 – x4 = f (x), the graph is
symmetrical about the y-axis.
4. Plot additional points and their reflections in the y-axis:
(1.5, 3.9) and (–1.5, 3.9 ), ( 0.5, 0.94 ) and (–0.5, 0.94)
y
5. Draw the graph.
(–1.5, 3.9 )
(–0.5, 0.94 )
(1.5, 3.9)
(0.5, 0.94)
x
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