Polynomial Functions
A Polynomial function of degree n is written is the form:
P  x   an x n  an 1 x n 1  an  2 x n  2 
 a2 x 2  a1 x  a0
where n is a non-negative integer exponent and a n  0
 n can only take on values where
 .a0 , a1 , a2 ,..., an 1 , an
 . an
 .a0
Example: For each of the following, determine if the given function is a polynomial. If it is a polynomial state
the degree and the leading coefficient.
f  x   8 x 4  3 x 2  12 x  4
g  x   3 x 3  4 x  10 x
h  x   2 x 6  3 x 3 
4
8
x2
j  x    x 2  3  2  5 x 
2
Graphing a polynomial function is a necessity in this course and all future courses. In calculus, you will learn that the
graph of a polynomial function is always a smooth curve (no sharp corners or cusps) and continuous (no holes or gaps).
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Polynomial Functions
Let's investigate the graph of the powers functions:
 Take the first few EVEN powers of n,
 (These are EVEN functions)
f  x   Ax n where n is any non-negative integer.
f  x   x2 , x4 , x6
* Behavior of the function about the x-intercept/Real zero:
* End Behavior of the function:
* (Basically, describe what y is doing as x grows without bound)
* What if the leading coefficient is NEGATIVE?
* (how will this affect the end behavior)
 Take the first few ODD powers of n,
 (These are ODD functions)
g  x   x3 , x5 , x7
* Behavior of the function about the x-intercept/Real zero:
* End Behavior of the function:
* (Basically, describe what y is doing as x grows without bound)
* What if the leading coefficient is NEGATIVE?
* (how will this affect the end behavior)
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Polynomial Functions
Sketch the function,
Sketch the function,
f  x   2  x3
g  x    x  3
based on the transformation of
6
based on the transformation of
x3
x6
SUMMARIZE THE END BEHAVIOR OF A POLYNOMIAL FUNCTION
 An nth degree polynomial function is only as good as its HIGHEST DEGREE TERM. That is, look at the degree of
the polynomial and the leading coefficient to determine the behavior of the polynomial as x gets large.
 That is, P  x   an x n  an 1 x n 1  an  2 x n  2 
 a2 x 2  a1 x  a0 behaves as the monomial, y  an x
 EVEN DEGREE
 ODD DEGREE
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n
Polynomial Functions
The Behavior of a Polynomial Function Near an x-intercept/Real zero.
*** The x-intercepts/Real zeroes play a major role in the graph of a polynomial function. At the x-intercepts/Real zeroes the
graph must either cross the x-axis or touch the x-axis.
Also, in between consecutive real zeroes the graph will either be above the x-axis or below the x-axis.
Example: Consider the following function is factored form:
f  x    x  3  x  2 
2
y-intercept:
Real Zero's: Set f(x) = 0
The function above is a perfect example of a 3rd degree polynomial with a positive leading coefficient.
The real zero of x = - 3 is said to have multiplicity of 1 while the real zero of x = 2 is said to have a
multiplicity of 2. Multiplicity refers to the number of times the zero occurs.
Example: Find a possible 4th degree polynomial whose real zero's are 3 with a multiplicity of 2,
and 2 and -2 both with a multiplicity of 1.
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Polynomial Functions
• Let f(x) be a polynomial function and suppose that
 x  a  is a factor of f(x). [Furthermore, assume that
none of the other factors of f(x) contains (x-a).] Then, in the immediate vicinity of the x-intercept/Real zero at a,
n
(here a is called a zero of multiplicity n) then the graph of y = f(x) closely resembles that of y  A  x  a 
n
• The principle that we haven just stated is easy to apply because we already know how to graph functions of
the form y  A  x  a  . The next example shows how this works.
f  x   x  x  2
2
 x  3
Real Zero's:
f  x   x  x  2   x  3
3
Real Zero's:
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Polynomial Functions
An Approach to Graphing Polynomial Functions
1. Factor the polynomial, if possible.
2. Find the y-intercept, set x = 0.
3. Find the real zeroes, set y = 0 and solve for x.
4. Determine the behavior near the real zero.
Even Multiplicity
graph touches (bounces) at the real zero.
-- Sign value of f(x) does NOT change on either side of the real zero.
Odd Multiplicity
graph crosses (cuts) at the real zero.
-- Sign value of f(x) CHANGES sign from one side of the real zero to the other.
5. Determine the end behavior.
Look at the degree of the polynomial and the leading coefficient to determine the behavior as x gets
large.
6. Plot points as necessary to determine the general shape of the graph. But, all we really need is one
point in any interval to determine whether the graph is above or below the x -axis in that interval, then
we can fill in the rest based on the knowledge of the multiplicity of the real zero.
Example: Sketch the function based on the above information:
f  x    x  2  x  3 
g  x   3 x  x  2   x  3 
3
Example: Sketch the function based on the above information:
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2
2
Polynomial Functions
Fun Fact: A polynomial function of degree n can have AT MOST n-1 turning points
Consider the third degree polynomial:
f  x   x3
g  x   x  x  3  x  4 
Example: Consider the graph of the following polynomial function f(x). Assume the polynomial has no
turning points beyond those shown.
Is the leading coefficient POSITIVE or NEGATIVE?
What is the minimum degree of this polynomial?
Find and describe the real zeros (and notice the multiplicity)
Find a possible polynomial that could model this graph if factored form.
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