1. 3.1 Symmetry and Coordinate Graphs
Objectives: Use algebraic tests to determine if the graph of a relation is symmetrical. Classify functions as even or odd.

2. Group work for lines of symmetry investigation
Two distinct points P and P’ are symmetric with respect to a line l iff. l is the bisector of PP’. A point P is symmetric to itself with respect to line l iff. P is on l.
*Can be folded in half on line of symmetry and two halves match exactly.
-common lines of symmetry are: x-axis, y-axis, y = x line and y = -x line. Pg
Line Symmetry:
Line l
P P’
y-axis symmetry

3. Symmetric with respect to: x-axis y-axis (also called even) y=x y = -x
(x,y) → (x,-y)
(x,y) → (-x, y)
(x,y) → (y, x)
(x,y) → (-y, -x)

4. Ex. 2:
Determine whether the graph of x² + y = 3 is symmetric with respect to the x-axis, the y-axis, the line y = x, y = -x or none of these.
Ex. 3)
Determine whether the graphs of y = x + 1
is symmetric with respect to the x-axis, the y-axis, both, or neither. Use the information to graph the relation.

5. Ex. 4)

6. Symmetric with respect to the origin (odd):
A function has a graph that is symmetric with respect to the origin iff. f(-x) = - f(x) for all x in the domain of f.
Symmetric with respect to the origin (odd):
*Looks the same upside down or right side up. A graph is symmetric with
respect to the origin if it is unchanged when reflected across both the x-axis and y-axis.
Ex. 1:
Determine whether each graph is symmetric with respect to the origin.
a.)
b.)
y = x²
c.)
g(x) = -3x³ + 5x

7. Even Functions:
Symmetric with respect to y-axis.
f(-x) = f(x)
Symmetric with respect to the origin.
f(-x) = -f(x)
Odd Functions: