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 real
numbers, an is not zero and is the leading coefficient, and the exponents are nonnegative integers.
The degree of a polynomial function is the highest degree exponent in the
polynomial.
Identifying Polynomial Functions
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 x 3 − 4 x + 8
2 4
x − 5 x3 − 12 x + 0.56
3
Characteristics:
Degree 0 – Constant
Degree 1 – Linear
Degree 2 – Quadratic
Degree 3 – Cubic
Degree 4 – Quartic
Degree 5 - Quintic
Odd Function
End behaviour
>0
End behaviour
<0
x–intercepts
y- intercepts
Domain
Range
Even Function
Every polynomial function that includes a variable has a value, or values, of the
variable for which the corresponding value of the function is zero. These values of
the variable are the zeroes of the function and are the solutions to the
corresponding equation f ( x) = 0 or y = 0 .
The real zeroes of a function are the x-intercepts of its graph.
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.
Match a Polynomial Function with its Graph
Ex. For each of the following functions, identify the following: the type of function
and whether it is of even or odd degree, the end behaviour of the graph of the
function, the number of possible x-intercepts, whether the graph will have a
maximum or minimum value, the y-intercept.
Then match each function to its corresponding graph.
)
c)
)=
)=−
+
−
+
−
−
−
b)
)=−
+
d)
)=
−
+
−
−
−
Answers:
a
b
c
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
6
8
4
6
2
4
x
0
-8
-6
-4
2
-6
-4
-2
0
2
4
6
8
-2
x
0
-8
-2
0
2
4
6
-4
8
-6
-2
-8
-4
-10
-6
-12
-14
-8
-16
-10
-18
-12
-20
-14
-22
-24
-16
C)
D)
y
y
28
26
4
24
22
20
18
16
2
14
12
10
8
x
0
-4
6
-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
d