1. Chapter 3 – The Nature of Graphs
Section 1 – Symmetry and Coordinate Graphs

2. Basic Symmetry
The most basic type of symmetry is point symmetry. A graph has point symmetry if it is symmetric about a SPECIFIC POINT.
For example, the origin (0,0) is a common point of symmetry.
f(x)=1/x and g(x)=x3 are both symmetric with respect to the origin.

4. Algebraically Determine Symmetry
To determine whether or not a function is symmetric with respect to the origin, we simply need to see if
–f(x)=f(-x).
If –f(x)=f(-x), then the function is symmetric about the origin.

5. Line Symmetry
If a function is symmetric with respect to the x-axis, then for some point (a,b) in the function, there should also be (a,-b) in the function.
If a function is symmetric with respect to the y-axis, then for some point (a,b) in the function, there should also be (-a,b) in the function.

6. More Line Symmetry
If a function is symmetric with respect to the line y=x, then for some point (a,b) in the function, there should be a point (b,a) also in the function.
If a function is symmetric with respect to the line y=-x, then for some point (a,b) in the function, there should be a point (-b,-a) also in the function.

7. SYMMETRY WITH RESPECT TO THE LINE
Line Symmetry Table
SYMMETRY WITH RESPECT TO THE LINE
DEFINITION
x-axis
If point (a,b) exists, then point (a,-b) must also exist.
y-axis
If (a,b) exists, then
(-a,b) must also exist.
y=x
If (a,b) exists, then (b,a) must also exist.
y=-x
(-b,-a) must also exist.

8. Figuring Lines of Symmetry
EX 1: Determine whether the graph of xy=-2 is symmetric with respect to the x-axis, y-axis, the line y=x, the line y=-x, or none of these.
First step, simply substitute (a,b) into our equation, and then substitute the rest of the points to see if they work out to be -2.

9. Even or Odd Functions
A function whose graph is symmetric with respect to the y-axis is an even function.
A function whose graph is symmetric with respect to the x-axis is an odd function.
It is possible for functions to be neither even nor odd.

10. Assignment
Chapter 3, Section 1
pgs #3,4,6-14,20,23,27,39, 42