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Polynomial Functions and Their Graphs
Definition of a Polynomial Function
Let n be a nonnegative integer and let an , an − 1 ,..., a2 , a1 , a 0 , be
real numbers, with an ≠ 0 .
The function defined by f ( x ) = an x n + ,...,+ a2 x 2 + a1 x + a0 is called
a polynomial function of x of degree n. The number an , the
coefficient of the variable to the highest power, is called the
leading coefficient.
Note: The variable is only raised to positive integer powers–no
negative or fractional exponents.
However, the coefficients may be any real numbers, including
fractions or irrational numbers like π or 7 .
Graph Properties of Polynomial Functions
Let P be any nth degree polynomial function with real coefficients.
The graph of P has the following properties.
1. P is continuous for all real numbers, so there are no breaks,
holes, jumps in the graph.
2. The graph of P is a smooth curve with rounded corners and no
sharp corners.
3. The graph of P has at most n x-intercepts.
4. The graph of P has at most n – 1 turning points.
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f (x ) = x( x − 2)3 ( x + 1)2
Polynomial
Not a polynomial
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Example 1: Given the following polynomial functions, state the
leading term, the degree of the polynomial and the leading
coefficient.
a. P( x ) = 7x 4 − 5 x 3 + x 2 − 7x + 6
b. P( x ) = (3x + 2)( x − 7)2 ( x + 2)3
End Behavior of a Polynomial
Odd-degree polynomials look like y = ± x 3 .
y = x3
y = −x3
Even-degree polynomials look like y = ± x2 .
y = x2
y = − x2
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Power functions:
A power function is a polynomial that takes the form f (x ) = axn ,
where n is a positive integer. Modifications of power functions can
be graphed using transformations.
Even-degree power functions:
f ( x) = x 4
Odd-degree power functions:
f (x) = x 5
Note: Multiplying any function by a will multiply all the y-values
by a. The general shape will stay the same. Exactly the same as it
was in section 3.4.
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Zeros of a Polynomial
Example 2:
Find the zeros of the polynomial and then sketch the graph.
P( x ) = x 3 − 5 x 2 + 6 x
If f is a polynomial and c is a real number for which f (c) = 0 , then
c is called a zero of f, or a root of f.
If c is a zero of f, then
• c is an x-intercept of the graph of f.
• (x − c ) is a factor of f.
So if we have a polynomial in factored form, we know all of its xintercepts.
• every factor gives us an x-intercept.
• every x-intercept gives us a factor.
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Example 3: Consider the function f (x ) = −3 x (x − 3)2 (4 − x )4 .
Zeros (x-intercepts):
To get the degree, add the multiplicities of all the factors:
The leading term is:
Steps to graphing other polynomials:
1. Factor and find x-intercepts.
2. Mark x-intercepts on x-axis.
3. Determine the leading term.
• Degree: is it odd or even?
• Sign: is the coefficient positive or negative?
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4. Determine the end behavior. What does it “look like”?
Odd Degree
Sign (+)
Even Degree
Sign (+)
Odd Degree
Sign (-)
Even Degree
Sign (-)
5. For each x-intercept, determine the behavior.
• Even multiplicity: touches x-axis, but doesn’t cross
(looks like a parabola there).
• Odd multiplicity of 1: crosses the x-axis (looks like a
line there).
• Odd multiplicity ≥ 3 : crosses the x-axis and looks like a
cubic there.
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Note: It helps to make a table as shown in the examples below.
6. Draw the graph, being careful to make a nice smooth
curve with no sharp corners.
Note: without calculus or plotting lots of points, we don’t have
enough information to know how high or how low the turning
points are.
Example 4:
Find the zeros then graph the polynomial. Be sure to label the x
intercepts, y intercept if possible and have correct end behavior.
P( x ) = x 4 (x − 2 )3 (x + 1)2
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Example 5:
Find the zeros then graph the polynomial. Be sure to label the x
intercepts, y intercept if possible and have correct end behavior.
P( x ) = x 3 (x + 2 )(x − 3)2
Example 6:
Find the zeros then graph the polynomial. Be sure to label the x
intercepts, y intercept if possible and have correct end behavior.
P( x ) = −2 (x + 1)2 (x − 7)4 (2 x − 10 )5
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Example 7:
Find the zeros then graph the polynomial. Be sure to label the x
intercepts, y intercept if possible and have correct end behavior.
P( x ) = x 3 + 3x 2 − 4 x − 12
Example 8:
Given the graph of a polynomial determine what the equation of
that polynomial.
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Example 9:
Given the graph of a polynomial determine what the equation of
that polynomial.