BGSU
2.3 Polynomial Functions and Their Graphs
Math 1300
Definition 1 (A Polynomial Function)
Let n be a nonnegative interger. Let a0 , a1 , . . . , an be real numbers and an 6= 0. Then we say that
f (x) = a0 + a1 x + . . . + an−1 xn−1 + an xn
is a polynomial of degree n. an is called the leading coefficient of f (x).
Good Properties of polynomial functions:
• Smooth: rounded curve without sharp corners
• Continuous: no breaks on the curve
End behavior of a polynomial (Leading Coefficient Test)
degree n
leading coefficient
left (x → −∞)
n is even
an > 0 an < 0
n is odd
an > 0 an < 0
right (x → ∞)
Example 1 Use the leading coefficient test to determine the end behavior of the give function.
a. f (x) = 3x4 − 999x3 + 1342x2 + 2x − 1
b. g(x) = −0.1(x − 3)(x + 2)2 x3
Zeros of Polynomials
Let f (x) be a polynomial. We say that a is a zero of f (x) if f (a) = 0.
Example 2 Find all zeros of the give function.
a. f (x) = −(x + 5)2 (x − 1)3
b. h(x) = x(x − 5) + 6
PS. More exercises can be found on P.331 on the textbook.
Ying-Ju Tessa Chen
Last modified: September 6, 2014
1
BGSU
2.3 Polynomial Functions and Their Graphs
Math 1300
Multiplicities of Zeros
If a polynomial f of degree n can be written by f (x) = (x − a)k Q(x) where Q(x) is a polynomial of degree
n − k and a is not a zero of Q(x). Then we say that a is a zero with multiplicity k.
Example 3 Find all zeros and their multiplicities of the give function. State whether the graph crosses
the x-axis or touches the x-axis and turns around at each zero.
a. f (x) = −x3 + 6x2 − 9x
b. g(x) = − 15 x(x + 12 )2 (x − 3)5
Remark 1 If a is a zero of even multiplicity, then the graph touches the x-axis and turns around at a.
If a is a zero of odd multiplicity, then the graph crosses the x-axis at a. If k is the multiplicity of a and
k > 1, then the graph of the function tends to flatten out near a. (Why?)
The Intermediate Value Theorem
Theorem 1 (The Intermediate Value Theorem for Polynomials)
Let f : R → R be a polynomial with real cofficients. If f (a)f (b) < 0, then there exists a point c between a
and b such that f (c) = 0.
Remark 2 In fact, the Intermediate Value Theorem holds for all continuous functions f : I → R where
I is a closed interval.
Ying-Ju Tessa Chen
Last modified: September 6, 2014
2
BGSU
2.3 Polynomial Functions and Their Graphs
Math 1300
Example 4 Use the Intermediate Value Theorem to show that each polynomial has a real zero between
the given integers.
a. f (x) = x3 − 3x2 + 1; between 0 and 1
b. g(x) = −0.5x4 − 2x3 + 3x2 ; between 1 and 2
Turning Points of Polynomials
If f is a polynomial of degree n, then the graph of f has at most n − 1 turing points. (Why?)
Graphing a polynomial
• Study the end behavior of the graph of the function by using the leading coefficient of the function.
• Find x-intercept by setting f (x) = 0 and y-intercept by setting x = 0 in the function f (x).
• Find all zeros and their multiplicities of the function.
• Check if the function is even or odd or neither.
• Use the fact ”If f is a polynomial of degree n, then the graph of f has at most n − 1 turning points.”
to check the graph.
Remark 3 In fact, the tool we learn here is not enough to sketch a ”correct” curve of a polynomial.
From the above steps, we may miss some turning points. But through the study here, we still get a picture
of how the graph looks like.
Ying-Ju Tessa Chen
Last modified: September 6, 2014
3