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Analyzing Graphs of Functions and Relations Unit 1 Lesson 2

Analyzing Graphs of Functions and Relations Unit 1 Lesson 2. Analyzing Graphs of Functions and Relations. Students will be able to : Analyze graphs of functions and relations ( x and y – intercepts, zeros, symmetry, even and odd functions) Key Vocabulary : Graph of a function,

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Analyzing Graphs of Functions and Relations Unit 1 Lesson 2

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  1. Analyzing Graphs of Functions and Relations Unit 1 Lesson 2

  2. Analyzing Graphs of Functions and Relations Students will be able to: Analyze graphs of functions and relations ( x and y – intercepts, zeros, symmetry, even and odd functions) Key Vocabulary: Graph of a function, An intercept, A zero of a function, Symmetry

  3. Analyzing Graphs of Functions and Relations • The graph of a function is the set of ordered pairs, in the coordinate plane, such that is the domain of . • the directed distance from the -axis • the directed distance from the -axis • You can use the graph to estimate function values.

  4. Analyzing Graphs of Functions and Relations Sample Problem 1: Use a graph of each function to estimate the indicated function values. Then find the values algebraically. a.

  5. Analyzing Graphs of Functions and Relations Sample Problem 1: Use a graph of each function to estimate the indicated function values. Then find the values algebraically. a.

  6. Analyzing Graphs of Functions and Relations Sample Problem 1: Use a graph of each function to estimate the indicated function values. Then find the values algebraically. a.

  7. Analyzing Graphs of Functions and Relations Sample Problem 1: Use a graph of each function to estimate the indicated function values. Then find the values algebraically. b.

  8. Analyzing Graphs of Functions and Relations Sample Problem 1: Use a graph of each function to estimate the indicated function values. Then find the values algebraically. b.

  9. Analyzing Graphs of Functions and Relations Sample Problem 1: Use a graph of each function to estimate the indicated function values. Then find the values algebraically. b.

  10. Analyzing Graphs of Functions and Relations Identifying Intercepts from a Functions Graph A point where the graph intersects or meets the or axis is called an intercept. An -intercept occurs where . A -intercept occurs where .

  11. Analyzing Graphs of Functions and Relations Sample Problem 2: Use the graph of each function to approximate its –intercept. Then find the –intercept algebraically. a.

  12. Analyzing Graphs of Functions and Relations Sample Problem 2: Use the graph of each function to approximate its –intercept. Then find the –intercept algebraically. a.

  13. Analyzing Graphs of Functions and Relations Sample Problem 2: Use the graph of each function to approximate its –intercept. Then find the –intercept algebraically. a. -intercept occurs where

  14. Analyzing Graphs of Functions and Relations Sample Problem 2: Use the graph of each function to approximate its –intercept. Then find the –intercept algebraically. b.

  15. Analyzing Graphs of Functions and Relations Sample Problem 2: Use the graph of each function to approximate its –intercept. Then find the –intercept algebraically. b.

  16. Analyzing Graphs of Functions and Relations Sample Problem 2: Use the graph of each function to approximate its –intercept. Then find the –intercept algebraically. b. -intercept occurs where

  17. Analyzing Graphs of Functions and Relations • Zeros of a Function • The zeros of function are –values for which • If the graph of a function of has an -intercept at then is a zero of the function. • To find the zeros of a function, set the function equal to zero and solve for the independent variable.

  18. Analyzing Graphs of Functions and Relations Sample Problem 3:Use the graph of each function to approximate its zeros. Then find the zeros of each function algebraically. a.

  19. Analyzing Graphs of Functions and Relations Sample Problem 3:Use the graph of each function to approximate its zeros. Then find the zeros of each function algebraically. a.

  20. Analyzing Graphs of Functions and Relations Sample Problem 3:Use the graph of each function to approximate its zeros. Then find the zeros of each function algebraically. a.

  21. Analyzing Graphs of Functions and Relations Sample Problem 3:Use the graph of each function to approximate its zeros. Then find the zeros of each function algebraically. b.

  22. Analyzing Graphs of Functions and Relations Sample Problem 3:Use the graph of each function to approximate its zeros. Then find the zeros of each function algebraically. b.

  23. Analyzing Graphs of Functions and Relations Sample Problem 3:Use the graph of each function to approximate its zeros. Then find the zeros of each function algebraically. b.

  24. Analyzing Graphs of Functions and Relations Symmetry of Graphs There are two possible types of symmetry that graphs of functions can have. 1. Line symmetry - graphs can be folded along a line so that the two halves match exactly. 2. Point symmetry - graphs can be rotated 180° with respect to a point and appear unchanged.

  25. Analyzing Graphs of Functions and Relations Tests for Symmetry

  26. Analyzing Graphs of Functions and Relations Sample Problem 4:Use the graph of each equation to test for symmetry with respect to the -axis, -axis, and the origin.Support the answer numerically. Then confirm algebraically. a.

  27. Analyzing Graphs of Functions and Relations Sample Problem 4:Use the graph of each equation to test for symmetry with respect to the -axis, -axis, and the origin.Support the answer numerically. Then confirm algebraically. a. Graphically The graph appears to be symmetric with respect to the origin because for every point (,) on the graph, there is a point (-,-).

  28. Analyzing Graphs of Functions and Relations Sample Problem 4:Use the graph of each equation to test for symmetry with respect to the -axis, -axis, and the origin.Support the answer numerically. Then confirm algebraically. Support Numerically a. There is a table of values to support this conjecture.

  29. Analyzing Graphs of Functions and Relations Sample Problem 4:Use the graph of each equation to test for symmetry with respect to the -axis, -axis, and the origin.Support the answer numerically. Then confirm algebraically. Algebraically a. • Because is equivalent to • the graph is symmetric with respect to the origin.

  30. Analyzing Graphs of Functions and Relations Sample Problem 4:Use the graph of each equation to test for symmetry with respect to the -axis, -axis, and the origin.Support the answer numerically. Then confirm algebraically. b.

  31. Analyzing Graphs of Functions and Relations Sample Problem 4:Use the graph of each equation to test for symmetry with respect to the -axis, -axis, and the origin.Support the answer numerically. Then confirm algebraically. b. Graphically The graph appears to be symmetric with respect to the x -axis because for every point (,) on the graph, there is a point (,-).

  32. Analyzing Graphs of Functions and Relations Sample Problem 4:Use the graph of each equation to test for symmetry with respect to the -axis, -axis, and the origin.Support the answer numerically. Then confirm algebraically. Support Numerically b. There is a table of values to support this conjecture.

  33. Analyzing Graphs of Functions and Relations Sample Problem 4:Use the graph of each equation to test for symmetry with respect to the -axis, -axis, and the origin.Support the answer numerically. Then confirm algebraically. Algebraically b. • Becauseis equivalent to • the graph is symmetric with respect to the .

  34. Analyzing Graphs of Functions and Relations • Identify Even and Odd Functions • If then the function is even, and symmetric to the y-axis. • If then the function is odd, and symmetric to the origin.

  35. Analyzing Graphs of Functions and Relations Sample Problem 5:Determine whether the following are even, odd, or neither. a.

  36. Analyzing Graphs of Functions and Relations Sample Problem 5:Determine whether the following are even, odd, or neither. a. The function is even.

  37. Analyzing Graphs of Functions and Relations Sample Problem 5:Determine whether the following are even, odd, or neither. b.

  38. Analyzing Graphs of Functions and Relations Sample Problem 5:Determine whether the following are even, odd, or neither. b. The function is odd.

  39. Analyzing Graphs of Functions and Relations Sample Problem 5:Determine whether the following are even, odd, or neither. c.

  40. Analyzing Graphs of Functions and Relations Sample Problem 5:Determine whether the following are even, odd, or neither. c. The function is neither.

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