1. 3-1 Symmetry & Coordinate Graphs Objective: 1. To determine symmetry of a graph using algebraic tests. 2. To determine if a function is even or odd.

2. Point Symmetry  Rotating a figure that has point symmetry 180 o should look the same.

3. Point Symmetry  2 points are symmetric with respect to (wrt) a 3 rd point “M” if & only if M is the midpoint of the 2 points. M

4. Point Symmetry  The graph of a relation S is symmetric WRT the origin if and only if (a,b)  S implies that (-a, -b)  S.

5. Point Symmetry  Steps for determining point symmetry about the origin: 1. Find f(-x) and –f(x) 2. If f(-x) =-f(x) then the graph has point symmetry.

6. Point Symmetry  Example: Determine whether the graph is symmetric WRT the origin. .

7. Determine whether each graph is symmetric WRT the origin a) b)

8. Line Symmetry  Two points P and P’ are symmetric wrt a line if and only if that line is the perpendicular bisector of PP’. A point P is symmetric to itself wrt a line if and only if P is on the line.  Four cases we will look at:  wrt x-axis  wrt y-axis  wrt y=x  Wrt y=-x

9. Line Symmetry wrt x-axis  If (a, -b)  S and (a, b)  S  Substituting (a, b) and (a, -b) into the equation produces the same result. 

10. Line Symmetry wrt x-axis  Example: x = y 2 (4,2), (4,-2) are both members of this set

11. Line Symmetry- wrt y-axis  If (-a, b)  S and (a, b)  S  Substituting (-a, b) and (a, b) into the equation produces the same result.

12. Line Symmetry- wrt y-axis  Example: y= x 2 -2 (-2, 2), (2, 2) are both members of this set

13. Line Symmetry- wrt y=x  If (b,a)  S and (a, b)  S  Substituting (b,a) and (a, b) into the equation produces the same result.

14. Line Symmetry- wrt y=x  Example: xy= 12 ( 2,6), (6,2) are both members of this set

15. Line Symmetry- wrt y=-x  If (-b,-a)  S and (a, b)  S  Substituting (-b,-a) and (a, b) into the equation produces the same result.

16. Line Symmetry- wrt y=-x  Example: xy= -12 ( 2,-6), (6,-2) are both members of this set

17. Practice  Determine whether the graph of x 2 + y = 3 is symmetric with respect to the x-axis, the y-axis the line y = x, the line y = -x.

18. Practice:  Can the graph of an equation be symmetric to more than one of these lines at a time? If yes, give an example, if no, give an explanation.

19. Practice  Determine whether the graph of |y|=|x|+1 is symmetric with respect to the x-axis, the y-axis, both or neither. Use the information about the equation’s symmetry to graph the relation.

20. Even and Odd Functions  Even functions have symmetry wrt the y-axis  Odd functions have symmetry wrt the origin

21. Practice:  Look at the previous examples from today and determine which are even or odd. even odd neither even odd