This document discusses properties of functions including whether they are even, odd, or neither based on their symmetry; whether they are increasing, decreasing, or constant over intervals; and how to identify local maxima and minima. Key aspects covered are that even functions are symmetric about the y-axis, odd functions are symmetric about the origin, and functions can be increasing, decreasing, or constant depending on whether the y-values increase, decrease, or remain the same as x-values change over an interval. Local extrema are also defined as maximums or minimums over open intervals.

1. 2.3 Properties of Functions Even, Odd, or Neither (Symmetry) Increasing and Decreasing Intervals Local Maxima and Minima

2. Even Function A function that is symmetric about the y-axis. Algebraically –

3. Odd Function A function that is symmetric about the origin (180⁰ rotational symmetry about the origin) Algebraically - same

4. Increasing As the x-values increase, the y-values also increase Describe the x-values in interval notation

5. Decreasing As the x-values increase, the y-values decrease Describe the x-values in interval notation

6. Constant As the x-values increase, the y-value remain the same. Describe the x-values in interval notation

7. Find the intercepts

8. State the Domain and Range

9. Identify the intervals where it is increasing Describe the x-values in interval notation

10. Identify the intervals where it is decreasing Describe the x-values in interval notation

11. Identify the intervals where it is constant None

12. Determine whether it is even, odd, or neither Symmetric about the origin – Odd Function

13. Local Maxima and Minima Local Maximum – The largest value of y on an open interval of x. Local Minima – The smallest value of y on an open interval of x.

14. Identify the local maximum The local maximum is 1, and it occurs when x = π/2

15. Identify the local minimum The local minimum is -1, and it occurs when x = -π/2

16. Identify the local extrema on the given interval (*using a calculator)

17. Find Maximum Estimate an interval of x 2nd TRACE (calc) 4:maximum (-2,0)

18. Find Maximum 2nd TRACE (calc) 4:maximum (-2,0)

19. Find Maximum -2 ENTER 0 ENTER ENTER (-2,0) Maximum is 11.53, occurs at x= -.82

20. Find Minimum Estimate an interval of x 2nd TRACE (calc) 3:minimum (0,2)

21. Find Minimum 0 ENTER 2 ENTER ENTER (0,2) Minimum is -1.53, occurs at x= .82

22. Assignment p. 88 # 7, 10 - 28 even, 39 - 46, 63, 65, 66