1. College Algebra Chapter 2 Functions and Graphs
Section 2.7
Analyzing Graphs of Functions and Piecewise-Defined Functions
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2. Concepts Test for Symmetry Identify Even and Odd Functions
Graph Piecewise-Defined Functions
Investigate Increasing, Decreasing, and Constant Behavior of a Function
Determine Relative Minima and Maxima of a Function

3. Concept 1 Test for Symmetry

4. Test for Symmetry
Consider an equation in the variables x and y. Symmetric with respect to the y-axis: Substituting –x for x results in equivalent equation. Symmetric with respect to the x-axis: Substituting –y for y results in equivalent equation. Symmetric with respect to the origin: Substituting –x for x and –y for y results in equivalent equation.

5. Example 1
Determine whether the graph of the equation is symmetric to the x-axis, y-axis, origin, or none of these.

6. Example 2
Determine whether the graph of the equation is symmetric to the x-axis, y-axis, origin, or none of these.

7. Example 3
Determine whether the graph of the equation is symmetric to the x-axis, y-axis, origin, or none of these.

8. Example 4
Determine whether the graph of the equation is symmetric to the x-axis, y-axis, origin, or none of these. Y=|x|-2 Solution: Y-axis : y =|-x|-2=|x|-2 yes X-axis : -y = |x|-2→ y=-|x|+2 No Origin :-y=|-x|-2 Y=-|x|+2 No

9. Skill Practice 1 (1 of 2)
Determine whether the graph is symmetric to the y-axis, x-axis, or origin.

10. Skill Practice 1 (2 of 2)
Determine whether the graph is symmetric to the y-axis, x-axis, or origin.

11. Concept 2 Identify Even and Odd Functions

12. Identify Even and Odd Functions
Even function: f(–x) = f(x) for all x in the domain of f. (Symmetric with respect to the y-axis) Odd function: f(–x) = –f(x) for all x in the domain of f. (Symmetric with respect to the origin)

13. Example 5
Determine if the function is even, odd, or neither.

14. Example 6
Determine if the function is even, odd, or neither.

15. Example 7
Determine if the function is even, odd, or neither.

16. Example 8
Determine if the function is even, odd, or neither.

17. Example 9
Determine if the function is even, odd, or neither.

18. Skill Practice 3 (1 of 2)
Determine if the function is even, odd , or neither. a.

19. Skill Practice 3 (2 of 2) b. c.

20. Skill Practice 4
Determine if the function is even, odd, or neither.

21. Concept 3 Graph Piecewise-Defined Functions

22. Example 10
Evaluate the function for the given values of x.

23. Example 11
Evaluate the function for the given values of x.

24. Skill Practice 5 Evaluate the function for the given values of x.

25. Example 12 (1 of 2) Graph the function. Y = - x - 1 x ≥ 2 x < 2 x y-2 1 2 x y 2 -3 3 -4 4 -5
x < 2
x ≥ 2

26. Example 12 (2 of 2)

27. Example 13 Y = - 1 Y =x - 1 Graph the function. Y =-2x + 6 x y -1 -2 x-1 -2 x y -1 1 2 x y 2 3 4 -2

28. Skill Practice 6
Graph the function.

29. Skill Practice 7
Graph the function.

30. Graph Piecewise-Defined Functions
Greatest integer function:
is the greatest integer less than or equal to x.

31. Example 14
Evaluate.

32. Example 15
Graph.

33. Skill Practice 8 Evaluate f(x)=[x] for the given values of x. f(1,7)

34. Example 16
A new job offer in sales promises a base salary of $3000 a month. Once the sales person reaches $50,000 in total sales, he earns his base salary plus a 4.3% commission on all sales of $50,000 or more. Write a piecewise-defined function (in dollars) to model the expected total monthly salary as a function of the amount of sales, x. Solution: When under m sales, only salary
$50,000+in sales means salary plus comm.

35. Skill Practice 9
A retail story buys T-shirts from the manufacturer. The cost is $7.99 per shirt for 1 to 100 shirts, inclusive. Then the price is decreased to $6.99 per shirt thereafter. Write a piecewise-defined function that expresses the cost C(x) (in $)to buy x shirts.

36. Concept 4 Investigate Increasing, Decreasing, and Constant Behavior of a Function

37. Investigate Increasing, Decreasing, and Constant Behavior of a Function (1 of 3)

38. Investigate Increasing, Decreasing, and Constant Behavior of a Function (2 of 3)

39. Investigate Increasing, Decreasing, and Constant Behavior of a Function (3 of 3)

40. Example 17
Use interval notation to write the interval(s) over which f(x) is increasing, decreasing, and constant.

41. Example 18
Use interval notation to write the interval(s) over which f(x) is increasing, decreasing, and constant. Increasing: never Decreasing: always (-∞,∞) Constant: never

42. Skill Practice 10
Use interval notation to write the interval(s) over which f is
Increasing
Decreasing
Constant

43. Concept 5 Determine Relative Minima and Maxima of a Function

44. Determine Relative Minima and Maxima of a Function

45. Example 19 (1 of 2)
Identify the location and value of any relative maxima or minima of the function. The point _(-2,-1)_ is the lowest point in a small interval surrounding x =_-2 . At x =_-2 the function has a relative minimum of _-1_. The point (0,1) is the highest point in a small interval surrounding x = _0_. At x =_0_ the function has a relative maximum of _1_.

46. Example 19 (2 of 2)

47. Example 20 (1 of 2)
Identify the location and value of any relative maxima or minima of the function. At x =_-1_ the function has a relative minimum of _-2__. At x = _4_the function has a relative minimum of _0_. At x =_2_ the function has a relative maximum of _3_.

48. Example 20 (2 of 2)

49. Example 21
Identify the location and value of any relative maxima or minima of the function. no max no min (inflection point)

50. Skill Practice 11 For the graph shown,
Determine the location and value of any relative maxima.
Determine the location and value of any relative minima.