1. 3.1 Symmetry and Coordinate Graphs
Students will use visual and algebraic tests to determine if graphs of relations are symmetric and classify functions as even or odd as evidenced by an exit slip.

2. What are the two main types of symmetry?
Point Symmetry
Line Symmetry
Two points, 𝑃 and 𝑃 ′ , have symmetry with respect to point 𝑀 if and only if 𝑀 is the midpoint of 𝑃𝑃′ .
Two points, 𝑃 and 𝑃 ′ , have symmetry with respect to line 𝑙 if and only if 𝑙 is the perpendicular bisector of 𝑃𝑃′ .

3. New symbols and concepts
𝑎, 𝑏 𝜖 𝑆, means the point (𝑎, 𝑏) is an element of the set S
2,4 𝜖 𝑥 2 since the point (2, 4) is on the graph of 𝑦= 𝑥 2
Function vs relation:
A function passes the vertical line test, a relation does not (think of a circle, we can still graph a circle, but its not a function)

4. Symmetry with Respect to the Origin (point symmetry)
The origin is the point ….?
Two ways of denoting origin symmetry
𝒂, 𝒃 𝜖 𝑆 implies −𝒂, −𝒃 𝜖 𝑆
E.g. 𝑥 3 , 𝟐, 𝟖 𝜖 𝑥 3 since −2, −8 𝜖 𝑥 3 we say that 𝑦= 𝑥 3 is symmetric with respect to the origin.
𝑓 −𝑥 =−𝑓 𝑥
Changing the sign of the INPUT changes the sign of the output.
This is also called an ODD function (not ODD exponent)

5. Symmetry with respect to a line (mirror)
There are four types of line symmetry
X-axis symmetry: 𝑎, 𝑏 , (𝑎, −𝑏)𝜖 𝑆
Y-axis symmetry: 𝑎, 𝑏 , (−𝑎, 𝑏)𝜖 𝑆
ALSO 𝑓 𝑥 =𝑓 −𝑥 , changing the sign of the input does not change the sign of the output.
This is also called an EVEN function.
𝑦=𝑥: 𝑎, 𝑏 , 𝑏,𝑎 𝜖 𝑆
𝑦=−𝑥: 𝑎, 𝑏 , −𝑏, −𝑎 𝜖 𝑆

6. Example 1: For the point (5, 2) give a point that has each type of symmetry, sketch a graph of each to verify.
( , ) has symmetry with respect to the origin.
( , ) has symmetry with respect to the line 𝑦=𝑥.
( , ) has symmetry with respect to the 𝑦−𝑎𝑥𝑖𝑠.
( , ) has symmetry with respect to the 𝑥−𝑎𝑥𝑖𝑠.
( , ) has symmetry with respect to the line 𝑦=−𝑥.
Origin (odd) : 𝑎, 𝑏 and −𝑎, −𝑏
𝑥-axis: 𝑎, 𝑏 and (𝑎, −𝑏)
𝑦-axis (even): 𝑎, 𝑏 and(−𝑎, 𝑏)
𝑦=𝑥: 𝑎, 𝑏 and 𝑏,𝑎
𝑦=−𝑥: 𝑎, 𝑏 and −𝑏, −𝑎

7. Example 2: For the point (−2, 4) give a point that has each type of symmetry
( , ) has symmetry with respect to the origin.
( , ) has symmetry with respect to the line 𝑦=𝑥.
( , ) has symmetry with respect to the 𝑦−𝑎𝑥𝑖𝑠.
( , ) has symmetry with respect to the 𝑥−𝑎𝑥𝑖𝑠.
( , ) has symmetry with respect to the line 𝑦=−𝑥.
Origin (odd): 𝑎, 𝑏 and −𝑎, −𝑏
𝑥-axis: 𝑎, 𝑏 and (𝑎, −𝑏)
𝑦-axis (even): 𝑎, 𝑏 and(−𝑎, 𝑏)
𝑦=𝑥: 𝑎, 𝑏 and 𝑏,𝑎
𝑦=−𝑥: 𝑎, 𝑏 and −𝑏, −𝑎

8. Example 3: Determine which types of symmetry the equation 𝑥𝑦=−2
Find a point on the graph (what times what will give you −2?)
Identify the point for each type of symmetry (think example 1 and 2).
Which points will WORK for the equation?
Name of Symmetry
Point with that symmetry
Origin (odd)
𝑥-axis
𝑦-axis (even)
𝑦=𝑥
𝑦=−𝑥
Name of Symmetry
Point with that symmetry
Will the new point work?
Origin (odd)
(−2, 1)
Yes
𝑥-axis
(2, 1) No
𝑦-axis (even)
−2, −1
𝑦=𝑥
(−1, 2)
𝑦=−𝑥
(1, −2)
Point that makes the equation TRUE:
2, −1
Does the graph validate your answers?
Think about the original problem/question. Write your solution in a complete sentence.

9. Example 4: Determine which types of symmetry the equation 𝑦= 𝑥 2 −𝑥
Find a point on the graph.
Identify the point for each type of symmetry.
Which points will WORK for the equation?
Name of Symmetry
Point with that symmetry
Origin (odd)
𝑥-axis
𝑦-axis (even)
𝑦=𝑥
𝑦=−𝑥
Name of Symmetry
Point with that symmetry
Will the new point work?
Origin (odd)
(−3, −6) No
𝑥-axis
(3, −6)
𝑦-axis (even)
(−3, 6)
𝑦=𝑥
(6, 3)
𝑦=−𝑥
(−6, −3) no
Point that makes the equation TRUE:
(3, 6)
Does the graph validate your answers?
Think about the original problem/question. Write your solution in a complete sentence.

10. Example 5: Determine which types of symmetry the equation 𝑥 2 = 5 𝑦 2
Find a point on the graph.
Identify the point for each type of symmetry.
Which points will WORK for the equation?
Name of Symmetry
Point with that symmetry
Will the new point work?
Origin (odd)
(− , −2)
Yes
𝑥-axis
( , −2)
𝑦-axis (even)
(− , 2)
𝑦=𝑥
(2, )
𝑦=−𝑥
(−2, − )
Name of Symmetry
Point with that symmetry
Origin (odd)
𝑥-axis
𝑦-axis (even)
𝑦=𝑥
𝑦=−𝑥
Point that makes the equation TRUE:
( , 2)
Does the graph validate your answers?
Think about the original problem/question. Write your solution in a complete sentence.

11. Summary
Sketch an example of each of the 5 types of symmetry. Be sure to label the type of symmetry.
Write a few sentences explaining to your friend who was absent today how to determine what type of symmetry an equation has. (not by looking at a picture, the EQUATION!)