1. 2.1 Symmetry

2. There are 3 symmetries important in graphing
When a figure has line symmetry, it can be folded on its line of symmetry and the two have to coincide exactly
There are 3 symmetries important in graphing
Respect to
x-axis
y-axis
origin
Definition
If (a, b) is on graph,
so is (a, –b)
so is (–a, b)
so is (–a, –b)
Illustration
Test
Substituting –y for y results in equivalent equation
Substituting –x for x results in equivalent equation
Substituting –x for x AND –y for y results in equivalent equation
(a, b)
(–a, b)
(a, b)
(a, b)
(a, –b)
(–a, –b)

3. Typically can do two types of problems
Ex 1) Given the point , determine coordinates of the point that satisfies specified symmetry.
origin
b) y-axis
c) x-axis
(a, b)  (–a, –b)
(a, b)  (–a, b)
(a, b)  (a, –b)

4. Ex 2) Determine symmetry. Graph using the properties of symmetry.
x-axis
–y = x2 – 4
y = –x2 + 4
not equivalent
y-axis
y = (–x)2 – 4
y = x2 – 4
equivalent
origin
–y = (–x)2 – 4
–y = x2 – 4
not equivalent x y 1 2 –4 –3 x y –1 –2 –3
Choose only positive x’s & negative x’s will match

5. Ex 3) Use graphing calculator to find symmetry.

6. We can also have graphs symmetric to line y = x
Test: switch x & y and get equivalent eqtn
(x, y)  (y, x)
Even Function
If f (–x) = f (x)
symmetric to y-axis
Odd Function
If f (–x) = – f (x)
symmetric to origin

7. Ex 4) Determine if function is even, odd or neithera)
odd b)
neither c)
even

8. Homework
# Pg #1–31 odd, 39, 41–46 all