“Why so serious?”
3.2 Polynomial Functions and Models
Polynomial Functions
A polynomial function is a function of the form
f (x)  an x  an1 x  ... a1 x  a0
where an ,an1,...,a1,a0 are real numbers and n is a
n
n1
nonnegative integer. The domain consists of all real numbers.


The degree of the function is the largest power of x.
Polynomial Functions
Determine which of the following are polynomial functions.
For those that are, state the degree.
f (x)  3  2x 5
g(x)  x  5
h(x)  3x 3 (x  2)2
3x 5
F(x) 
5  2x
G(x)  6
Polynomial Functions
Determine which of the following are polynomial functions.
For those that are, state the degree.
4 x 3  3x  5
f (x) 
2
g(x)  7x(2x  3) 4 (x  5) 2
h(x)  x  ex  2.634
3
2
F(x)  x  2x  8
4
G(x)  x  2x  3
e
Power Functions
A power function is of the form
f (x)  x
If n is odd…
n
If n is even…

The end behavior of a polynomial function can be related to
the power function of the same degree.
Zeroes of Polynomials
Find the degree, the end behavior, zeroes, and multiplicity each.
f (x)  x 2 (x  3)(x  2) 3

If a zero has an odd multiplicity, the function crosses at the zero.
If a zero has an even multiplicity, the function touches at the zero.
Zeroes of Polynomials
Find the degree, the end behavior, zeroes, and multiplicity each.
f ( x)   x3 ( x  2) 4 ( x  8)
Zeroes of Polynomials
Find the polynomial whose zeroes are given.
1) Zeroes: 0, -4, 2 (all multiplicity 1)
2) Zeroes: -3, multiplicity 2; 5, multiplicity 1
3) Zeroes: 0, multiplicity 2; 6, multiplicity 3; -2, multiplicity 4
Zeroes of Polynomials
Given the polynomial f (x)  x(x  4) 2 (x  3)(x 2  1)
a) Find the degree and end behavior of the polynomial.
b) Find the x-and y-intercepts of the graph.
c) Determine whether the graph crosses or touches at each xintercept.
d) Graph f using a graphing utility.
e) Determine any local maxima or minima.
Zeroes of Polynomials
Given the polynomial f (x)  x(x  4) 2 (x  3)(x 2  1)
f) Graph by hand.
g) Determine the domain and range.

h) Determine intervals of increasing
and decreasing
Degree = 6 y-intercept: (0,0)
x-intercepts: (0,0) crosses, (4,0) touches; (-3,0) crosses
Local minima: (-2.321,-402.176) and (4,0)
Local maxima: (2.451,224.636)
Zeroes of Polynomials
Construct a polynomial function that might have this graph.
3.2 Polynomial Functions and Models
Zeroes of Polynomials
a) Find the degree and end behavior of the polynomial.
b) Find the x- and y-intercepts of the graph.
c) Determine whether the graph crosses or touches at each xintercept.
d) Graph f using a graphing utility.
e) Determine any local maxima or minima.
f) Graph by hand. (accurately plot all known points!)
g) Determine the domain and range.
h) Determine intervals of increasing
and decreasing