SWBAT identify polynomials and their degree
(Lesson 4 - Section 3-2 and 3-3)
Warm up
1) The function given by 𝑓(𝑥) = −12𝑥 2 − 1 has no intercepts. True or False
2) The graphs of 𝑓(𝑥) = −4𝑥 2 − 10𝑥 + 7 and 𝑓(𝑥) = 12𝑥 2 + 30𝑥 + 1 have the same axis of
symmetry. True or False.
Definition of a Polynomial Function
Let n be a nonnegative integer and let 𝑎𝑛, 𝑎𝑛−1, … … … … 𝑎2, 𝑎1, 𝑎 0 be real numbers
with 𝑎𝑛, ≠ 0. The function given by
𝑓(𝑥) = 𝑎𝑛, 𝑥 𝑛 + 𝑎𝑛,−1 𝑥 𝑛−1 + … . . + 𝑎2 𝑥 2 + 𝑎 1 𝑥 + 𝑎0
is called a polynomial function of x with degree n.
A polynomial function is a function whose rule is given by a polynomial in one
variable.
The degree of a polynomial function is the largest power of x that appears.
The zero polynomial function f(x) = 0 + 0x + 0x 2 +… +0x n is not assigned a
degree.
Identifying Polynomial Functions
Example 1) Determine which of the following are polynomial functions. For
those that are, state the degree; for those that are not, tell why not.
x
a) f(x) = 2 - 3x 4
b) g(x) =
x2  2
c) h(x) = 3
x 1
d) f(x) = 0
e) f(x) = 8
f) f(x) = - 2x 3 (x - 1) 2
Graphs of Polynomial Functions
In this section, you will study basic feature of the graphs of polynomial functions.
1) The graph of a polynomial function is continuous. This means that the graph has no
breaks, holes, or gaps.
2) The graph of a polynomial function has only smooth, rounded turns. A polynomial
function cannot have a sharp turn.
Power Functions
The polynomial functions that have the simplest graphs are monomials of the form
𝑓 (𝑥) = 𝑥 𝑛 , where n is an integer greater than zero.
When n is even, the graph is similar to the graph of 𝑓 (𝑥) = 𝑥 2 , and when n is odd, the
graph is similar to the graph 𝑓(𝑥) = 𝑥 3 .

The greater the value of n, the flatter the graph is near the origin. Polynomial
functions of the form 𝑓 (𝑥) = 𝑥 𝑛 are often referred to as power functions.
Example 1) Comparing Graphs
b)
Practice
(a) f  x   3x 5  4 x 4  2 x 3  5
1
3
(b) g  x   3x 2  5 x  10
(c) h  x   3x  5
(d) F  x   2 x 3  3x  8
(e) G  x   5
(f) H  s   3s  2s 2  1