Section 3.2 – Polynomial Functions and their graphs
math 130
Definition
1. Polynomial functions
a) Coefficient
b) Leading term
c) Leading coefficient
d) Degree
Graphs of Polynomials
A graph of a polynomial function will always be continuous. …(NO BREAKS). We already know how to
graph TWO types of polynomials … polynomials of degree 1, polynomials of degree 2. With the help of
this knowledge and transformations we can also ‘sketch’ other polynomial functions.
2. Polynomial behavior
a) If n is odd AND the leading coefficient a n , is positive, the graph falls to the left and rises to
the right
b) If n is odd AND the leading coefficient a n , is negative, the graph rises to the left and falls to
the right.
c) If n is even AND the leading coefficient a n , is positive, the graph rises to the left and to the
right.
d) If n is even AND the leading coefficient a n , is negative, the graph falls to the left and to the
right.
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with a positive
leading coefficient
with a negative
leading coefficient
with a positive
leading coefficient
with a negative
leading coefficient
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x such that f(x) = 0. In
other words it is the x-intercept, where the functional value or y is equal to 0.
3. A zero or root of a polynomial function is the value of
4. Multiplicity of Zeros
a) Odd multiplicity
b) Even multiplicity
5. Intermediate Value Theorem
Let a and b be two numbers with a<b. If f is a polynomial function such that f a  and f b  have
opposite signs, then there is at least one number c with a  c  b , for which f c   0
Example: Use the Intermediate Value Theorem to show that each polynomial has a real zero in the
specified interval. Approximate this zero to two decimal places.
Px   x 4  x 2  2 x  5 ; [1,2]
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Example #1 – Graph the polynomial px   x 4  2 x 3  3x 2
Step 1: Determine the graph’s end behavior. Use the Leading Coefficient test described above to find if
the graph rises or falls to the left and to the right
Since the degree of the polynomial, 4, is even and the leading coefficient, 1, is
positive, then the graph of the given polynomial
rises to the left and rises to the right.
Step 2: Find the x-intercepts or ZEROS of the function.
 0  x 4  2 x 3  3x 2
Step 3: Determine the multiplicity and whether the graph CROSSES or TOUCHES the x-axis.
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

X=0 has multiplicity 2. This means the graph touches the x axis and turns around.
X=3 has multiplicity 1. This means the graph ______________________________
X=-1 has multiplicity 1. This means the graph ______________________________
Step 4: Find the y-intercepts of the function
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Example #2
Answer the following questions for the function
Px   2 x 3  x 2  x
a) Describe the end behavior of the function
b) Find the zeros and multiplicities
c) Find the the y-intercepts
d) Use the information on parts a-c to sketch your function. You may need to extra points
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Example #3
Consider the function Px   3x 4  18x 3  27 x 2
a) Describe the end behavior of the function
b) Find the zeros and multiplicities
c) Find the the y-intercepts
Use the information on parts a-c to sketch your function. You may need to extra points.
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Example #4
Consider the function Px   x 5  9x 3
a) Describe the end behavior of the function
b) Find the zeros and multiplicities
c) Find the the y-intercepts
d) Use the information on parts a-c to sketch your function. You may need to extra points.
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