Transcript Graphs of Polynomial Functions

Graphs of Polynomial
Functions
What You Should Learn
(do not write down)
• Determine key features of a polynomial graph
• Use the Leading Coefficient Test to determine
the end behavior of graphs of polynomial
functions.
• Find and use zeros of polynomial functions as
sketching aids.
• Find a polynomial equation given the zeros of the
function.
Polynomials
• What do you remember about polynomials??
• What would be key points of a polynomial?
• Remember this …
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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.
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A polynomial function is a function of the form below….
Polynomial Function
Leading Coefficient
Degree
f ( x)  2 x5  3x3  5 x  1
-2
5
f ( x)  x 3  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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Leading Coefficient Test
• Odd Degree:
• Leading coefficient is positive, then the graph’s end
behaviors will be opposite.
• Even Degree:
• Leading coefficient is negative, then the graph’s end
behavior will be the same.
y
y
an positive
x
x
n odd
an negative
n even
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Find the left and right behavior of the
polynomial.
1. y  x  3 x  2 x
5
3
2. f ( x )   x  4 x  1
4
3
3. y  2 x  3 x  1
2
5
4. f ( x )  3 x  2 x  8
4
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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.)
Given the polynomials below, answer the following
A. What is the degree?
B. What is its leading coefficient?
C. How many “turns”(relative maximums or minimums) could it have (maximum)?
D. How many real zeros could it have (maximum)?
E. How would you describe the left and right behavior of the graph of the equation?
F. What are its intercepts ?
Equations:
1. y   x 3  3 x 2  2 x
2. f ( x )  x 4  2 x 2  8
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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.
Example – Solution
cont’d
Solution:
To find the real zeros of the function, set f(x) equal to zero
and solve for x.
–2x4 + 2x2 = 0
–2x2(x2 – 1) = 0
Remove common monomial factor.
–2x2(x – 1)(x + 1) = 0
Factor completely.
Set f(x) equal to 0.
So, the real zeros are x = 0 (double root), x = 1, and x = –1.
Because the function is a fourth-degree polynomial, the graph of f can
have at most 4 – 1 = 3 turning points.
Zeros of Polynomial Functions
In the example, note that because the exponent
is greater than 1, the factor –2x2 yields the
repeated zero x = 0.
Because the exponent is even,
the graph touches the
x-axis at x = 0.
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: Sketch the graph of f (x) = 4x2 – x4.
1. Write the polynomial function in standard form: f (x) = –x4 + 4x2
The leading coefficient is negative and the degree is even.
as x  , f (x )  
2. Find the zeros of the polynomial by factoring.
f (x) = –x4 + 4x2 = –x2(x2 – 4) = – x2(x + 2)(x –2)
Zeros:
x = –2, 2 multiplicity 1
x = 0 multiplicity 2
y
(–2, 0)
(2, 0)
x
(0, 0)
x-intercepts:
(–2, 0), (2, 0) crosses through
(0, 0)
touches only
Example continued
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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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Find the Polynomial
Given the zeros, find an equation (assume lowest degree):
Zeros: 2, 3
Answer: (x – 2)(x – 3) = x2 – 5x +6
Zeros: 0 (multiplicity of 2), -2, 5
Answer: x2 (x + 2)(x – 5)= x2(x2 – 3x + 10)
= x4 – 3x3 + 10x2
Zeros: 2, 3 (multiplicity of 2), -4(multiplicity of 3) – leave in
factored form
Answer: (x – 2)(x – 3)2 (x + 4)3
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