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3. MATHEMATICAL REASONING INSTITUTE LESSON GOALS 3  A.5.e – For a function that models a linear or nonlinear relationship between two quantities, interpret key features of graphs and tables in terms of quantities, and sketch graphs showing key features of graphs and tables in terms of quantities, and sketch graphs showing key features given a verbal description of the relationship.  Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior, and periodicity.

4. MATHEMATICAL REASONING INSTITUTE x-intercept y-intercept Positive interval Negative interval Decreasing interval Increasing interval Relative maximum Relative minimum End behavior Line symmetry Rotational symmetry Periodicity Key Features of Graphs 4

5. MATHEMATICAL REASONING INSTITUTE Identifying Key Features of a Linear Graph X –intercept = -1.5 Y-intercept = 3 Has positive slope, thus it is increasing. Positive interval: graph is above the x-axis when x>-1.5 Negative interval: graph is below the x-axis when x<-1.5

6. MATHEMATICAL REASONING INSTITUTE Identifying Key Features of a Graph 6 Identify the following: x-intercept y-intercept Positive interval Negative interval y = | x |

7. MATHEMATICAL REASONING INSTITUTE Identifying Key Features of a Graph 7 Identify the following: Decreasing interval Increasing interval Decreasing interval: graph falls from left to right when x<0 Increasing interval: graph rises from left to right when x>0 y = | x |

8. Key Features of a Quadratic Function Graph https://www.youtube.com/watch ?v=H46QVUdbBn0 8

9. MATHEMATICAL REASONING INSTITUTE Identifying Key Features from a Graph Identify the following: x-intercept y-intercept Positive interval Negative interval Decreasing interval Increasing interval Relative maximum Relative minimum

10. MATHEMATICAL REASONING INSTITUTE Identifying Key Features from a Graph Identify the following: Relative maximum Relative minimum The y-coordinate of any point that is the highest/lowest point for some section of the graph.

11. End Behavior of Graphs https://www.youtube.com/watch ?v=PbSJHr-fg7I 11

12. MATHEMATICAL REASONING INSTITUTE End Behavior of Graphs 12 As x→+∞, f(x)→+∞ As x →-∞, f(x) →-∞

13. MATHEMATICAL REASONING INSTITUTE Graphs that have line symmetry have two halves that are mirror images of each other. Graphs that have rotational symmetry can be rotated around a point to coincide with itself. Some graphs just do neither and have no symmetry. Symmetry of Graphs 13

14. MATHEMATICAL REASONING INSTITUTE Line Symmetry 14

15. MATHEMATICAL REASONING INSTITUTE Rotational Symmetry 15

16. MATHEMATICAL REASONING INSTITUTE What Kind of Symmetry? 16

17. MATHEMATICAL REASONING INSTITUTE Periodic Graphs 17

18. MATHEMATICAL REASONING INSTITUTE LESSON GOALS 18  A.5.e – For a function that models a linear or nonlinear relationship between two quantities, interpret key features of graphs and tables in terms of quantities, and sketch graphs showing key features of graphs and tables in terms of quantities, and sketch graphs showing key features given a verbal description of the relationship.  Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior, and periodicity.

19. MATHEMATICAL REASONING INSTITUTE Sketch a graph with the following key features: y-intercept = -3 x-intercepts = -3, -1, 3, and 5 There is one relative minimum, -4, occurring at one point. There is one relative maximum, 1, occurring at two points. The graph is symmetric over the line x=1. Sketching Graphs using Key Features 19

20. MATHEMATICAL REASONING INSTITUTE Small Group Exercise! Common Core Achieve Exercise Book pp. 67-68, #’s 1-11

21. Lunch Break! Please come back on time. 21