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Maximum, Minimum, Even, and Odd

Maximum, Minimum, Even, and Odd. NYOS Charter School Precalculus. Nonlinear Functions and Their Graphs. An Absolute (Global) Maximum is the maximum y-value across the entire domain of a function. An Absolute (Global) Minimum is the minimum y-value across the entire domain of a function.

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Maximum, Minimum, Even, and Odd

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  1. Maximum, Minimum, Even, and Odd NYOS Charter School Precalculus

  2. Nonlinear Functions and Their Graphs An Absolute (Global) Maximum is the maximum y-value across the entire domain of a function. An Absolute (Global) Minimum is the minimum y-value across the entire domain of a function.

  3. Nonlinear Functions and Their Graphs An Local (Relative) Maximum is the maximum y-value within a restricted domain of a function. An Local (Relative) Minimum is the minimum y-value within a restricted domain of a function.

  4. Nonlinear Functions and Their Graphs Example: What are the local and global extrema? Local Maximum: 10 Local Minimum: -10 Global Maximum: none Global Minimum: none

  5. Nonlinear Functions and Their Graphs Example: What are the local and global extrema? Local Maximum: Local Minimum: Global Maximum: Global Minimum:

  6. Nonlinear Functions and Their Graphs Example: What are the local and global extrema? Local Maximum: 1, 3 Local Minimum: -2, 0 Global Maximum: 3 Global Minimum: none

  7. Nonlinear Functions and Their Graphs Assignment Draw a graph that has at least three local maximums and three local minimums.

  8. Nonlinear Functions and Their Graphs • A function is an even function if f(-x) = f(x) for every x in the domain. • The graph is symmetric with respect to the y-axis.

  9. Nonlinear Functions and Their Graphs Example: Is the function f(x) = x2 – 10 even? f(-x) = (-x)2 – 10 = x2 – 10 which is f(x). Thus, f(x) is even.

  10. Nonlinear Functions and Their Graphs Example: Is the function f(x) = 2x4 + 21 even? f(-x) =

  11. Nonlinear Functions and Their Graphs Example: Is the function f(x) = 2x4 + 21 even? f(-x) = 2(-x)4 + 21 = 2x4 + 21 which is f(x). Thus, f(x) is even.

  12. Nonlinear Functions and Their Graphs Example: Is the function f(x) = x3 – 5 even? f(-x) =

  13. Nonlinear Functions and Their Graphs Example: Is the function f(x) = x3 – 5 even? f(-x) = (-x)3 – 5 = -x3– 5 which is not f(x). Thus, f(x) is not even.

  14. Nonlinear Functions and Their Graphs Example: Is the function represented in the graph even?

  15. Nonlinear Functions and Their Graphs Example: Is the function represented in the graph even? The graph is symmetric with respect to the y-axis. Thus, f(x) is even.

  16. Nonlinear Functions and Their Graphs • A function is an odd function if f(-x) = -f(x) for every x in the domain. • The graph is symmetric with respect to the origin.

  17. Nonlinear Functions and Their Graphs Example: Is the function f(x) = 5x odd? f(-x) = 5(-x) = -5x which is –f(x). Thus, f(x) is odd.

  18. Nonlinear Functions and Their Graphs Example: Is the function f(x) = -x5 odd? f(-x) =

  19. Nonlinear Functions and Their Graphs Example: Is the function f(x) = -x5 odd? f(-x) = -(-x)5 = x5 which is -f(x). Thus, f(x) is odd.

  20. Nonlinear Functions and Their Graphs Example: Is the function represented by the graph odd? which is f(x).

  21. Nonlinear Functions and Their Graphs Example: Is the function represented by the graph odd? which is f(x). The graph is symmetric with respect to the origin. Thus, f(x) is odd.

  22. Nonlinear Functions and Their Graphs Example: Is the function f(x) = -x6 + 5x2 even or odd? f(-x) =

  23. Nonlinear Functions and Their Graphs Example: Is the function f(x) = -x6 + 5x2even or odd? f(-x) = -(-x)6 + 5(-x)2 = -x6 + 5x2 which is f(x). Thus, f(x) is even.

  24. Nonlinear Functions and Their Graphs Example: Is the function f(x) = x + 3 even, odd, or neither? f(-x) =

  25. Nonlinear Functions and Their Graphs Example: Is the function f(x) = x + 3 even, odd, or neither? f(-x) = -x + 3 which is neither f(x) nor -f(x). Thus, f(x) is neither.

  26. Nonlinear Functions and Their Graphs Assignment: • Draw a graph that has at least three local maximums and three local minimums. • Create three functions • One of each: even, odd, neither • Must contain at least three terms each.

  27. Nonlinear Functions and Their Graphs • Visit Khanacademy.org • Set up an account if you don’t have one • Be sure to add me as your coach! ttacker@nyos.org • Complete the “Even and Odd Functions” exercise

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