Download Graphs of Polynomial Functions

Warm up
Determine algebraically whether each of the following functions is even, odd or
neither.
Write the equation for transformation of.
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Pre-Calculus
Graphs of Polynomial
Functions
What You Should Learn
• 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. 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 x5  3x3  5x  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
4
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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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.)
Finding the zeros of polynomial functions is one of the most
important problems in algebra.
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 (y for all, x for 1 & 2 only)?
1. y   x 3  3 x 2  2 x
Equations:
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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Zeros of Polynomial Functions
• There is a strong interplay between graphical and
algebraic approaches to this problem.
• Sometimes you can use information about the graph of
a function to help find its zeros, and in other cases you
can use information about the zeros of a function to
help sketch its graph.
• Finding zeros of polynomial functions is closely related
to factoring and finding x-intercepts.
A real number a is a zero of a function y = f (x)
if and only if f (a) = 0.
Real Zeros of Polynomial Functions
If y = f (x) is a polynomial function and a is a real number then
the following statements are equivalent.
1. a is a zero of f.
2. a is a solution of the polynomial equation f (x) = 0.
3. x – a is a factor of the polynomial f (x).
4. (a, 0) is an x-intercept of the graph of y = f (x).
A turning point of a graph of a function is a point at which the
graph changes from increasing to decreasing or vice versa.
A polynomial function of degree n has at most n – 1 turning
points and at most n zeros.
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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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Zeros of Polynomial Functions
This means that when the real zeros of a polynomial
function are put in order, they divide the real number
line into intervals in which the function has no sign
changes.
These resulting intervals are test intervals in which a
representative x-value in the interval is chosen to
determine if the value of the polynomial function is
positive (the graph lies above the x-axis) or negative
(the graph lies below the x-axis).
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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Can you???
• 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.
Homework 8
• 2.2 Page 130
– 1-8 all (matching)
– 13-18(left and right behavior), all
– 27-41 odds (finding zeros-verify with a calculator)
– 47-55 odds
• Quiz next class – Complex Numbers
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