POLYNOMIAL FUNCTIONS
(An Introduction)
A polynomial function has the form:
f(x) = anxn + an-1xn-1 + an-2xn-2 + … + a2x2 + a1x + a0
where
a0, a1, … , an are called the coefficients and a  R
a0 is the constant term
an is the leading coefficient
x is the variable
n is the degree of the polynomial and n  W
(NOTE: a polynomial function contains only 1 variable with powers arranged in descending order)
What does a polynomial function look like algebraically?
Polynomial Functions
NOT Polynomial Functions
𝑦=𝑥
𝑦 = 2𝑥 −3
𝑓(𝑥) = 𝑥 2 + 3𝑥 − 5
𝑦 = 𝑥𝑦 + 3𝑥 − 4
𝑦=
√5 3
𝑥 − 2√5𝑥
2
𝑓(𝑥) = 4
𝑦 = (𝑥 − 1)3 (2𝑥 + 5)5
𝑓(𝑥) =
WHY????
1
𝑥
𝑦 = 3𝑥 3 − 10𝑥 1.5
𝑦 = √𝑥
What does a polynomial function look like graphically?
Some typical examples are shown below:
The shape of a polynomial function’s graph depends on the degree of the function.
Ex 
Decide which of the following graphs represent polynomial functions:
A.
Ex 
B.
C.
D.
Use finite differences to determine if the given set of data is linear or quadratic:
Time (s)
Height (m)
0
1
2
3
4
21
13
7
3
1
First
Difference
Second
Difference
What type of function has constant third differences? Fourth differences?
SUMMARY
 A polynomial function must be in one variable & the exponents must be whole numbers.
 The degree of the function is the highest exponent in the expression.
 The domain of any polynomial function is the set of all real numbers, D = { 𝑥 𝜖 𝑅 }.
 If the degree of the function is odd, the range is the set of all real numbers, R = { 𝑦 𝜖 𝑅 }.
 If the degree of the function is even, the range will have an upper or lower
bound (but not both).
 The graphs of polynomial functions do not have asymptotes.
 A polynomial of degree n, has constant nth finite differences.
Ex. f(x) = 6x3 – 2x2 + x – 5 has a degree of 3 and has constant third differences
Homework: p.127–128 #1–3, 5, 7, 8