Day 3: Characteristics of Polynomial Functions
A polynomial function is an equation of the form
f ( x)  an x n  an 1 x n 1              a1 x 1  a0
where the coefficients an , an1 ,........, a1 , a0 represent _________ numbers, an (known as the
________________ ______________) is not zero, and the exponents are ____-____________ integers.
The ____________ of a polynomial function is the _____________ degree exponent in the polynomial.
Ex: Determine whether each function is a polynomial function. If it is not, explain why, if it is a
polynomial, identify the leading coefficient, the degree, and the constant term.
b) y  3x 2  4 x 2  6
a) y  3x  11
d) y  5  4 x 
1
x
e) y 
c) y  2 x3  4 x  8
2 4
x  5 x3  12 x  0.56
3
Graphs of Polynomial Functions
Degree 0 – Constant
Degree 1 – Linear
Degree 2 – Quadratic
Degree 3 – Cubic
Degree 5 - Quintic
Degree 4 – Quartic
Characteristics of Polynomial Functions
Odd Function
End behaviour 𝑎𝑛 > 0
End behaviour 𝑎𝑛 < 0
x–intercepts
y- intercepts
Domain
Range
Even Function
Some polynomial functions ________ the x-axis. When the polynomial crosses the x-axis, the values
of the x variables are known as the _____________ of the function and are the solutions to the
corresponding equation f ( x)  0 or y  0 .
The ________ _______________of a function are the x-intercepts of its graph.
Additional Notes: Some graphs of polynomial functions have symmetry. The function y  x3 is
symmetric about the origin (a 180° rotation maps y  x3 onto itself). The function y  x 2  1 is
symmetric about the y-axis.
Ex: Complete the table below for the following functions. Then match each function to its
corresponding graph.
𝒂) 𝒈(𝒙) = 𝒙𝟒 + 𝟒𝒙𝟑 − 𝒙𝟐 − 𝟏𝟔𝒙 − 𝟏𝟐
b) 𝒇(𝒙) = −𝒙𝟒 + 𝟗𝒙𝟐 + 𝟓𝒙 − 𝟒
c) 𝒌(𝒙) = −𝟐𝒙𝟓 + 𝟓𝒙𝟑 − 𝟐𝒙
d) 𝒉(𝒙) = 𝟑𝒙𝟑 − 𝒙𝟐 − 𝟒𝒙 − 𝟐
Answers:
a
b
c
d
Type
Odd or Even
End Behaviour
Maximum number of x-intercepts
Maximum or minimum
y -intercept
Graphs:
A)
B)
16
16
y
y
14
14
12
12
10
10
8
8
6
4
6
2
x
4
0
-8
-6
-4
2
x
-6
-4
-2
0
2
4
6
8
-2
-4
0
-8
-2
0
2
4
6
8
-6
-2
-8
-4
-10
-6
-12
-14
-8
-16
-10
-18
-12
-20
-22
-14
-24
-16
C)
D)
y
y
28
26
4
24
22
20
18
16
2
14
12
10
8
x
0
6
-4
-2
0
4
2
x
0
-8
-6
-4
-2
0
-2
2
4
6
8
-2
-4
-6
-8
-10
-4
-12
-14
2
4