Section 2.2 More on Functions and Their Graphs

1. Section 2.2 More on Functions and Their Graphs

2. Increasing and Decreasing Functions

4. The open intervals describing where functions increase, decrease, or are constant, use x-coordinates and not the y-coordinates.

5. Use the graph to determine the intervals on which • the function is increasing. • (-4,-2), (0,2), (4,5) c) (4,-4), (4,0) • (-5,-4), (-2,0), (2,4) d) (0,4), (-4,4)

6. Example Find where the graph is increasing? Where is it decreasing? Where is it constant?

7. Example Find where the graph is increasing? Where is it decreasing? Where is it constant?

8. Relative Maxima And Relative Minima

9. The points at which a function changes its increasing or decreasing behavior can be used to find the relativemaximum or relative minimumvalues of the function. Page 217

10. Notice thatfdoes not have a relative maximum or minimum at - and , the x-intercepts, or zeros, of the function. Page 218

11. Example Where are the relative minimums? Where are the relative maximums?

12. Why are the maximums and minimums called relative or local? • The word local is sometimes used instead of relative • when describing maxima or minima. • If f has a relative, or local, maximum at a, f(a) is greater than all other values of f near a. • If f has a relative, or local, maximum at b, f(b) is less • than all other values of f near b.

13. Even and Odd Functions and Symmetry

15. A graph is symmetric with respect to the y-axis if, for every point (x, y) on the graph, the point (-x, y) is also on the graph. • A graph is symmetric with respect to the origin if, for every point (x, y) on the graph, the point (-x, -y) is also on the graph

16. Example Determine whether each function is even, odd, or neither.

17. Example Is this an even or odd function?

18. A function that is defined by two (or more) equations over a specified domain is called a piecewise function. Piecewise Functions

20. Example Find and interpret each of the following.

21. Evaluate the piecewise function at the given values of the independent variable.

22. Example Graph the following piecewise function.

23. Functions and Difference Quotients

24. See next slide.

25. Example 6 Find and simplify the expressions if

26. Example Find and simplify the expressions if

27. Example Find and simplify the expressions if

29. Look at the table and the accompanying graph. • int(x) = the greatest integer that is less than or equal to x

30. (a) (b) (c) (d)

31. (a) (b) (c) (d)

32. (a) (b) (c) (d)