2.3 Properties of Functions
A heart is not judged by how much you love, but by how much you
are loved by others.
-Wizard of Oz
Even and Odd Functions
A function is even if for every number x in its domain, the
number -x is also in its domain and
f (x)  f (x)
Ex : f (x)  x 2  2
Symmetry about
the y-axis

A function is odd if for every number x in its domain, the
number -x is also in its domain and
2x 3
Ex : f (x) 
3

f (x)   f (x)
Symmetry about
the origin
Even and Odd Functions
State whether each function is even, odd, or neither. Then
state any symmetry.
Even and Odd Functions
Determine whether the following functions are even, odd, or
neither. Then state any symmetry.
1) f (x)  x 4  x 2  4
2x
2) f (x) 
x
5
f
(x)

4
x
 3x
3)
4) f (x)  x 2  3x  1
Decreasing, Increasing, and Constant (OH MY!)
Determine the intervals when the following function is decreasing,
increasing, or constant.
Local Minima and Maxima
Local Minima and Maxima
Locate any local minima and maxima in the following graph.
Graphs
Use the graph of the following function identify any local maxima
or minima. Find all intervals where the function is increasing,
decreasing and constant.
y
4
a) Local Maxima
(2, 3)
b) Local Minima
(4, 0)
0
(1, 0)
-4
(0, -3)
x
(10, -3)
c) Increasing
d) Decreasing
(7, -3)
e) Constant
Average Rate of Change
Given a function y = f (x), the average rate of change of f from a
to b is defined as
y f (b)  f (a)
Average rate of change 

x
ba
Secant Line
(b, f (b))

f (b) – f (a)
(a, f (a))

b–a

a
b
Average Rate of Change
Given a function y = f (x), the average rate of change of f from a
to b is defined as
y f (b)  f (a)
Average rate of change 

x
ba
Secant Line
(16,7)

7– 3
(5,3)

16 – 5

5
1
6
Average Rate of Change
Find the average rate of change of the function below from -2 to 6.
Then find the equation of the secant line connecting the points.
h(x)  x 2  2x  3

Average rate of change 
y f (b)  f (a)

x
ba
Homework: p.89 #11-20 all,
2.3 Properties of Functions #21- 27 odd, #33-43 odd,
#53-59 odd
A heart is not judged by how much you love, but by how much you
are loved by others.
-Wizard of Oz