Avon High School
Section: 3.2
ACE COLLEGE ALGEBRA II - NOTES
Polynomial Functions and Their Graphs
Mr. Record: Room ALC-129
Day 1 of 1
Polynomial Functions
Definition of a Polynomial Function
Let n be a nonnegative integer and let a1 , an1 ,
The function defined by
, a2 , a1 , a0 be real numbers, with an  0.
is called a polynomial function of degree n. The number an , the coefficient of the
variable to the highest power, is called the leading coefficient.
Investigation: Which of the following are polynomial functions and which are not?
3
h( x )  2  2 x 2  5
f ( x)  3x5  2 x 2  5
g ( x)  3 x  2 x 2  5
k ( x)  3x 4 ( x  2)( x  3)
x
Smooth, Continuous Curves
Polynomial functions of degree 2 or higher have graphs that are smooth and continuous. By smooth, we mean
that the graphs contain only rounded curves with no sharp corners. By continuous, we mean that the graphs
have no breaks and can be drawn without lifting your pencil from the rectangular coordinate system. These two
ideas are illustrated below.
End Behavior of Polynomial Functions
The behavior of a graph of a function to the far left or the far right is called its end behavior. Although the
graph of a polynomial function may have intervals where it increased or decreases, the graph will eventually
rise or fall without bound as it moves far to the left or far to the right.
The best way to determine this behavior is using the Leading Coefficient Test.
The Leading Coefficient Test
As x increases or decreased without bound, the graph of the polynomial function
f ( x)  an x n  an 1 x n 1   a1 x  a0 (a  0)
eventually rises or falls. In particular,
1. For n odd
2. For n even
If the leading coefficient
is positive, the graph
falls to the left and rises
to the right. (↓,↑)
Example 1
If the leading coefficient
is negative, the graph
rises to the left and falls
to the right. (↑,↓)
If the leading coefficient
is positive, the graph
rises to the left and rises
to the right. (↑,↑)
If the leading coefficient
is negative, the graph
falls to the left and falls
to the right. (↓,↓)
Using the Leading Coefficient Test
Use the Leading Coefficient Test to determine the end behavior of the graph of
a. f ( x)  4 x3 ( x  2)2 ( x  5)
b.
f ( x)  2 x3 ( x  1)( x  5)
1.
2.
3.
4.
5.
Determine whether the parabola opens upward or downward by determining the value of a.
Determine the vertex,
, of the parabola.
Find any x-intercepts by solving
. The function’s real zeros are the x-intercepts.
Find the y-intercepts by computing
Plot the intercepts, the vertex, and additional points as necessary using a chart.
Connect these points with a smooth curve that is shaped like a bowl or inverted bowl.
Using the Leading Coefficient Test
Example 2
Sketch the graph of f ( x)   x 4  8x3  4 x 2  2 with a graphing calculator on view it on the
default window. Does this graph illustrate the proper end behavior? Why or why not?
is in standard form. The graph of f is a parabola whose vertex is at
. The parabola is
symmetric with respect to the line
. If
, the parabola opens upward; if
, the
parabola opens downward.
Zeros of Polynomial Functions
Example 3
Finding Zeros of a Polynomial Function
Find all the zeros of f ( x)  x3  2 x 2  4 x  8 . Sketch the function on a graphing calculator
and verify you’re the zeros you found analytically.
Multiplicities of Zeros
Example 4
Finding Zeros and Their Multiplicities
2
1

Find all the zeros of f ( x)  4( x  2)  x   ( x  5)3 and give the multiplicity of each
2

zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at
each zero. Verify your conclusions by using a graphing calculator.
Multiplicity and x-Intercepts
If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is
a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether the
multiplicity of a zero is even or odd, graphs tend to flatten out near zeros with multiplicity greater
than one.
The Intermediate Value Theorem
The Intermediate Value Theorem
Let f be a polynomial function with real coefficients. If f
(a) and f (b) have opposite signs, then there exists at least
one value of c between a and b for which f (c)  0 .
Equivalently, the equation f ( x)  0 has at least one real
root between a and b.
Example 5
Using the Intermediate Value Theorem
Show that the polynomial function f ( x)  3x3  10 x  9 has a real zero between -3 and -2.
Turning Points of Polynomial Functions
At each of the turning points for the function shown to the right, the graph
changes from increasing to decreasing or vice versa. The function illustrated
here has 5 as its greatest exponent and is therefore a polynomial function of
degree 5. Notice the graph has 4 turning points. If f is a polynomial
function of degree n, then the graph of f has at most n-1 turning points.
A Strategy for Graphing Polynomial Equations
While nothing can compare to using a graphing calculator, the following guidelines will help you graph
polynomials without the use of technology, while reinforcing some of the most important concepts we’ve
discussed over the past few days.
Graphing a Polynomial Function
f ( x)  an x n  an 1 x n 1  an  2 x n  2  a1 x  a0 , an  0
1. Use the Leading Coefficient Test to determine the graph’s end behavior.
2. Find x-intercepts by setting f ( x)  0 and solving the resulting polynomial
equation. If there is an x-intercept at r as a result of ( x  r ) k in the complex
factorization of f ( x) , then
a. If k is even, the graph touches the x-axis and turns around.
b. If k is odd, the graph crosses the x-axis at r.
c. If k  1 , the graph flattens out near (r , 0)
3. Find the y-intercept by computing f (0) .
4. Use symmetry, if applicable, to help draw the graph:
a. y-axis symmetry: f ( x)  f ( x)
b. origin symmetry: f ( x)   f ( x)
5. Use the fact that the maximum number of turning points of the graph is n - 1, where
n is the degree of the polynomial function, to check whether it is drawn correctly.
Let f be a polynomial function with real coefficients. If f (a) and f (b) have opposite
signs, then there exists at least one value of c between a and b for which
.
Example 6
Graphing a Polynomial Function
Graph f ( x)  x3  3x 2
y
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