1. FUNCTIONS
Symmetric about the y axis
Symmetric about the origin

2. 2-7 -6 -5 -4 -3 -2 -1 1 5 730 4 6 8
7
1
2
3
4
5
6
8
-2
-3
-4
-5
-6
-7
So for an even function, for every point (x, y) on
the graph, the point (-x, y) is also on the graph.
Even functions have y-axis Symmetry

3. 2-7 -6 -5 -4 -3 -2 -1 1 5 730 4 6 8
7
1
2
3
4
5
6
8
-2
-3
-4
-5
-6
-7
So for an odd function, for every point (x, y) on the
graph, the point (-x, -y) is also on the graph.
Odd functions have origin Symmetry

4. 2-7 -6 -5 -4 -3 -2 -1 1 5 730 4 6 8
7
1
2
3
4
5
6
8
-2
-3
-4
-5
-6
-7
We wouldn’t talk about a function with x-axis symmetry
because it wouldn’t BE a function.
x-axis Symmetry

5. A function is even if f( -x) = f(x) for every number x in
the domain.
So if you plug a –x into the function and you get the
original function back again it is even.
( ) 125 24
+−= xxxf Is this function even?
( ) 1251)(2)(5 2424
+−=+−−−=− xxxxxf
YES
( ) xxxf −= 3
2 Is this function even?
( ) xxxxxf +−=−−−=− 33
2)()(2
NO

6. A function is odd if f( -x) = - f(x) for every number x in
the domain.
So if you plug a –x into the function and you get the
negative of the function back again (all terms change signs)
it is odd.
( ) 125 24
+−= xxxf Is this function odd?
( ) 1251)(2)(5 2424
+−=+−−−=− xxxxxf
NO
( ) xxxf −= 3
2 Is this function odd?
( ) xxxxxf +−=−−−=− 33
2)()(2
YES

7. If a function is not even or odd we just say neither
(meaning neither even nor odd)
( ) 15 3
−= xxf
Determine if the following functions are even, odd or
neither.
( ) ( ) 1515 33
−−=−−=− xxxf
Not the original and all
terms didn’t change
signs, so NEITHER.
( ) 23 24
+−−= xxxf
( ) 232)()(3 2424
+−−=+−−−−=− xxxxxf
Got f(x) back so
EVEN.

8. Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au