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4.1
Polynomial Functions
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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 , a0 , be real numbers, with an  0 .
The function defined by f ( x)  a n x n ,..., a 2 x 2  a1 x  a 0 is called a polynomial function of x
of degree n. The number a n , 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  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.
2.
3.
4.
P is continuous for all real numbers, so there are no breaks, holes, jumps in the graph.
The graph of P is a smooth curve with rounded corners and no sharp corners.
The graph of P has at most n x-intercepts.
The graph of P has at most n – 1 turning points.
f ( x)  x( x  2)3 ( x  1) 2
M 1310
4.1
Polynomial Functions
Polynomial
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Not a polynomial
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  5x 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   x3 .
y  x3
y  x 3
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4.1
Polynomial Functions
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Even-degree polynomials look like y   x 2 .
y  x2
y  x 2
Power functions:
A power function is a polynomial that takes the form f ( x)  ax n , 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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4.1
Zeros of a Polynomial
Polynomial Functions
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Example 2:
Find the zeros of the polynomial and then sketch the graph.
P( x)  x 3  5x 2  6x
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 x-intercepts.
 every factor gives us an x-intercept.
 every x-intercept gives us a factor.
Example 3: Consider the function f (x)  3x(x  3)2 (4  x) 4 .
Zeros (x-intercepts):
To get the degree, add the multiplicities of all the factors:
The leading term is:
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4.1
Polynomial Functions
Steps to graphing other polynomials:
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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?
4. Determine the end behavior. What does it “look like”?
Odd Degree
Sign (+)
Odd Degree
Sign (-)
Even 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.
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.
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4.1
Polynomial Functions
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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
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
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Example 6:
4.1
Polynomial Functions
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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 2x  105
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  4x  12
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Polynomial Functions
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Example 8:
Given the graph of a polynomial determine what the equation of that polynomial.
Example 9:
Given the graph of a polynomial determine what the equation of that polynomial.