1. Hope your Weekend was Relaxing!  Pick up notes from the front table  Pick up new assignment log  Begin your Entry Ticket  Tonight’s HW: o Pg. 133 #2,3 15-29 odd, 39 AND draw a picture of a real life object that has symmetry. Identify the point or line of symmetry please use graph paper. o YOU NEED your graphing calculator for next class period or you will not be able to participate in the activities and thus you will lose participation points!  Updates: o Unit 3 Quiz 1 (3.1-3.2) Thursday/Friday

2. Congratulations! You made it through the first two units!!! From now on, I am going to print your guided notes for you

3. Agenda Entry Ticket! (Return quizzes) 3.1: Symmetry and Coordinate Graphs Function Flash! Cool-Down…

4. Notes/Homework Log I am missing several homework logs and notes from students. This is effecting your grade. Please turn those in!

5. Return Work! I am returning all work that I have of yours. o The Unit 1 Projects will be handed back the Monday we get back from break. ( graphing animals) o Unit 2 Assessment will be returned Thursday/Friday ( hopefully!)

6. Entry Ticket

7. Learning Objectives ① Use algebraic tests to determine if the graph of a relation is symmetrical. ② Classify functions as even or odds.

8. 3.1 Symmetry and Coordinate Graphs Unit 3 is all about the nature of graphs. To help us sketch and analyze graphs in this unit we need to understand types of symmetry.

9. 3.1 Symmetry and Coordinate Graphs Point Symmetry o Two distinct points P and P’ are symmetric with respect to the point of symmetry if (1) the two distinct points are the same distance from the point of symmetry AND (2) they are opposite direction from one another.

10. 3.1 Symmetry and Coordinate Graphs In your tables:  Identify the point of symmetry for each figure.  Use evidence to prove to me that the King demonstrates point symmetry.  How many degrees will I have to rotate to be symmetric about a point?

11. 3.1 Symmetry and Coordinate Graphs Point Symmetry o Two distinct points P and P’ are symmetric with respect to the point of symmetry if (1) the two distinct points are the same distance from the point of symmetry AND (2) they are opposite direction from one another. o A figure that is symmetric with respect to a given point is rotated 180°.

12. 3.1 Symmetry and Coordinate Graphs Whiteboards: Using the definition of Point Symmetry, identify if the following demonstrate point symmetry. If it does, provide the exact point. (1) (2) (3)

13. 3.1 Symmetry and Coordinate Graphs Notice that both whiteboard (1) and (2) were symmetric about (0,0) or the origin. In your tables discuss the following: o Specifically looking at the table, what do you notice about the points when they are symmetric about the origin?

14. 3.1 Symmetry and Coordinate Graphs Symmetry with Respect to the Origin If a graph is symmetric about the origin, it will demonstrate the relationship f(-x) = - f(x).

15. Examples/ Non-Examples

16. 3.1 Symmetry and Coordinate Graphs Attempt Practice (1) using the definition of symmetry about the origin.

17. 3.1 Symmetry and Coordinate Graphs We have discussed two types of symmetries so far: ① Point Symmetry ② Origin Symmetry Now, let’s discuss line symmetry! On your whiteboards, draw an example of line symmetry.

18. 3.1 Symmetry and Coordinate Graphs Line Symmetry o Two distinct points P and P’ are symmetric with respect to a line if the two points are reflected (or flipped) about the line. o Figures or graphs that have line symmetry can be folded along the line of symmetry so that the two halves match exactly. o Some graphs have more than one line of symmetry.

19. 3.1 Symmetry and Coordinate Graphs Common line symmetries we see in graphs are with respect to the x-axis, y-axis, y = x, and y = -x. Whiteboards: In your tables see if you can predict where the point (2, 4) will be when symmetric with respect to: x-axis y-axis y = x y = -x Hint: Keep in mind that symmetry must be the same distance from the pre-image to the line and from the image to the line.

20. 3.1 Symmetry and Coordinate Graphs Symmetry with Respect to: x – axis: Point P(a, b) image would be P’(a, -b). y – axis: Point P(a, b) image would be P’(-a, b). y = x: Point P(a, b) image would be P’(b, a). y = -x: Point P(a, b) image would be P’(-b, -a).

21. Reminder: Absolute value: - x = x

22. 3.1 Symmetry and Coordinate Graphs Let’s attempt (2a) together and then I am going to challenge you with (2b). You can use your graphing calculator to look at the function visually!

23. 3.1 Symmetry and Coordinate Graphs Common names you heard with symmetry in Algebra 2 were even and odd. o In your tables, discuss what you remember about even and odd functions.

24. 3.1 Symmetry and Coordinate Graphs Even Functions o Symmetric with respect to the y-axis. o f(-x) = f(x) 0 Maps (x, y)  (-x, y) 0 Graph will match exactly if you fold “hotdog” style ( over the y- axis) Real life logo:

25. 3.1 Symmetry and Coordinate Graphs Odd Functions o Symmetric with respect to the origin. o Maps (x, y)  (-x, -y) o f(-x) = -f(x) 0 Graph will match exactly if you fold the graph along the y-axis (fold hot dog style). Then fold along the x-axis (hamburger style). 0 It does NOT matter if you do hamburger first or hotdog first. Real-life picture:

26. Neither Odd nor Even Functions 0 Any function that isn’t even or odd

27. Part of Your Homework… Is to draw a real life object and point out it’s symmetry so we are going to practice with real life pictures. You may NOT use any of the following pictures that I put on this PowerPoint for your homework. BE CREATIVE! For the following pictures identify the point or line of symmetry. There may be none.

28. Part of Your Homework… Is to draw a real life object and point out it’s symmetry so we are going to practice with real life pictures. You may NOT use any of the following pictures that I put on this PowerPoint for your homework. BE CREATIVE! For the following pictures identify the point or line of symmetry. There may be none

29. Odd and Even Functions 0 Take out your whiteboards and whiteboard pens. 0 I am going to display a graph and you are going to write on your whiteboard ODD, EVEN, or NEITHER. 0 ARE you READY? 0 GET SET!!

30. Odd and Even Functions 0 GO!!

31. Odd and Even Functions

39. Answer the following on your whiteboards: You are given an ordered pair and you are to tell me where that ordered pair will map to given it is part of an even function AND given when it is part of an odd function. (a) (5, 2) (b) (-4, 4) (c) (0, -3)

40. Whiteboards Algebraically show that the following functions are even, odd, or neither. (a) h(x) = x 3 – 1 (b) F(x) = 3x 3 (c) d(x) = 2x 3 – x (d) k(x) = -3x 4 – x 2 + 2