2. Ch 2 Quarter TEST Review

3. RELATION A correspondence between 2 sets …say you have a set x and a set y, then… x corresponds to y y depends on x x is the input and y is the output x y

4. Ways to express a relation: Equation Table Graph Mapping 0 1 1 3

5. FUNCTION A relation from set x into set y that associates with element of x with exactly one element of y. inputs MUST have only ONE output outputs MAY be repeated for different inputs

6. Example 1A Tell if the relation is a function; if so state domain and range Mom Dad Kari Seth 555-2341 555-7890 555-8541 555-2222 555-3213 555-8504 NOT A FUNCTION!!!

7. Example 1B Tell if each relation is a function; if so state domain and range Hamburger Cheese- burger Chef Salad $4.00 $4.50 $5.00 YES a function!!! Domain: {Hamburger, Cheeseburger, Chef Salad} Range: {$4.00, $4.50, $5.00}

8. Example 1C Tell if each relation is a function; if so state domain and range YES a function!!! Domain:{410, 580, 750, 600, 430}Range: {19, 29, 33, 23} Even though 23 occurs twice, just list it once in the range…

9. {(1,4), (3,5), (5,7), (7,9)} YES a function!! Domain: {1,3,5,7}Range: {4,5,7,9} Example 1D Tell if each relation is a function; if so state domain and range

10. {(4,-2), (1,-1), (0,0), (1,1), (4,2)} NOT a function!! Because the 4 and 1 in the domain each have more than one output! Example 1E Tell if each relation is a function; if so state domain and range

11. {(-2,4), (-1,1), (0,0), (1,1), (2,4)} Outputs CAN be repeated…just not elements of the domain! Example 1F Tell if each relation is a function; if so state domain and range YES a function!! Domain: {-2,-1,0,1,2} Range: {4,1,0}

12. So, if we have a table of values or a mapping, we can tell whether or not our relation is a function. But, what about equations? (1) Solve for y (put the function in its explicit form) (2) Check if any value in the domain will generate more than one value for y. a) Look at the table, or b) Look at the graph (vertical line test)

13. Example A Step 1: solve for y by adding 3x to each side… Example B Step 1: solve for y by subtracting x 2 from both sides… Step 2: graph the two equations on calculator… Example 2 Determine whether the equation defines y as a function of x. Step 2: graph on calculator/look at table… Take the square root: +3x -x 2-x 2 -x 2-x 2

14. Finding Values of a Function f(x) means “the value of f at the number x” The independent variable is called the argument x is the input and f(x) is the output or value…

15. Example 3: Evaluate the following for the function, f, defined by Just plug it in & simplify! A. f(3) = 3(3) 2 + 5(3) =42 B. f(x) + f(3) = f(x) + 42 Remember: f(3) = 42… = 3x 2 + 5x + 42 …and f(x) is 3x 2 + 5x = 3x 2 + 5x + 42

16. Example 3 cont… C. f(-x) = 3(-x) 2 + 5(-x) D. - f(x) = - (3x 2 + 5x) This means the opposite of f(x)… = -3x 2 – 5x …distribute the negative… (-x) 2 =x 2 = 3x 2 + 5(-x) 5(-x) = -5x = 3x 2 - 5x = - (3x 2 + 5x)

17. Example 3 cont… E. f(x + 3) = 3(x + 3) 2 + 5(x + 3) (x + 3) 2 = (x + 3)(x + 3) = x 2 +6x + 9 = 3(x 2 + 6x + 9) + 5(x + 3) distribute the 3 and 5… = 3x 2 + 27 + 18x + 5x + 15 …combine like terms = 3x 2 + 23x + 42

18. Example 3 cont… F. f(x + h) = 3(x + h) 2 + 5(x + h) = 3(x 2 + 2xh + h 2 ) + 5(x + h) distribute the 3 and 5… = 3x 2 + 6xh+ 5x+ 3h 2 + 5h 1 st take care of f(x + h) f(x + h) = 3x 2 + 6xh + 3h 2 + 5x + 5h

19. Example 3F cont… f(x + h) – f(x) = = 3x 2 + 6xh + 3h 2 + 5x + 5h - (3x 2 + 5x) distribute the negative… = 3x 2 + 6xh + 3h 2 + 5x + 5h …combine like terms… = 6xh + 3h 2 + 5h = h (6x + 3h + 5) …and factor. = 6x + 3h + 5 - 3x 2 - 5x

20. If an equation for a function, f, is given with no specified domain, it is agreed that the domain of f is the largest set of real numbers for which the value of f(x) is a real number. The Domain of a Function In other words: The domain is all inputs that make sense and give an answer for the equation…

21. Example 4: Find the domain of each function. A. What can x NOT be?? Verify with a graph! all real numbers B. Cannot have a zero on the bottom! x is undefined at -1 and 1 x can be anything!

22. Example 4 continued: Find the domain of each function. C. You cannot take the square root of a negative number! …add 5t to both sides… +5t …divide both sides by 5… or

23. Functions can be added, subtracted, multiplied and divided: Operations on Functions If f and g are functions, then...

24. f+g: (3x + 7) + (2x - 4) = 5x+3 f-g: (3x + 7) - (2x - 4) = 3x + 7 – 2x + 4 = x+11 Example 5A: Given f(x) = 3x + 7 and g(x) = 2x – 4, find f + g, f - g, f * g, and f /g. (combine like terms) (distribute negative sign) (combine like terms)

25. f *g: (3x + 7)(2x - 4) Example 5A continued: Given f(x) = 3x + 7 and g(x) = 2x – 4 = 6x 2 + 2x - 28 (FOIL...multiply) f/g: (3x + 7)/(2x – 4) (simplify) = 6x 2 – 12x+ 14x– 28 (There is nothing more to do!) = 3x + 7 2x - 4

26. f+g:f+g: Example 5B: Given f(x) = 3/x and g(x) = 1/x, find f + g, f - g, f * g, and f /g. (must have a common denominator whenever adding or subtracting fractions) f-g:f-g:

27. f *g: Example 5B continued: Given f(x) = 3/x and g(x) = 1/x (top x top bottom x bottom) f/g:f/g: (flip and multiply!)

28. Vertical Line Test A set of points in the xy-plane is the graph of a function iff (if and only if) every vertical line intersects the graph in at most one point. Example: Determine whether the graph represents a function. No Yes

29. Many things can be learned about a function from its graph, such as (but certainly not limited to) values of the function (f(x)) at various values of x, the domain and range, the intercepts (both x and y), the number of times a function is intersected by other functions, and values of x that generate different function values.

30. EXAMPLE (a)Is the point (1, ½) on the graph of f? (use table on calc.) yes (b)If x = 2, what is f(x)? What point is on the graph of f? f(x)= ⅔ (2, ⅔ ) (c)If f(x) = 2, what is x? What point is on the graph of f? x = -2 (-2,2) (d)Find any x-intercepts. (0, 0) (a)Find any y-intercepts. (0, 0) (e)

31. EXAMPLE The average cost C of manufacturing x computers per day is given by the function Determine the average cost of manufacturing: (a) 30 computers in a day (b) 40 computers in a day (c) 50 computers in a day $1351.54 $1232.97 $1293.07

32. What do the y-values need to be?

33. What value of x minimizes the average cost? Use TRACE to find approximate minimum, then TABLE to verify Producing 41 computers a day will minimize the cost to $1231.75.

34. Determining Even and Odd Functions from the Graph A function, f, is even if for every number x in its domain the number –x is also in the domain and f(-x) = f(x)...that is, if (x, y) is on the graph, then (-x, y) is too. Also, a function is even iff its graph is symmetric with respect to the y-axis. f (-x) = f (x)

35. An EVEN Function f (-x) = f (x) (-2, 2) & (2, 2) Graph must go in same direction and be symmetric to the y-axis!

36. Determining Even and Odd Functions from the Graph A function, f, is odd if for every number x in its domain the number –x is also in the domain and f(-x) = - f(x)...that is, if (x, y) is on the graph, then (-x, -y) is too. Also, a function is odd iff its graph is symmetric with respect to the origin. f (-x) = -f (x)

37. An ODD Function f (-x) = -f (x) (-4, 8) & (4, -8) Graph must go in opposite directions and intersect the origin!

38. Example: Determine whether each graph given is an even function, an odd function, or neither even nor odd. Even OddNeither

39. Identifying Even and Odd Functions Algebraically EXAMPLE Determine whether each function is even or odd...verify algebraically. A. To verify algebraically, you must prove that f(-x) = f(x) Since f(-x) = f(x), then it’s an EVEN function! Graph it…does it appear to be even or odd?

40. Identifying Even and Odd Functions Algebraically B. To verify algebraically, first prove that Graph it…does it appear to be even or odd? Then it’s NEITHER even nor odd! Therefore it is not even… So now prove that Not odd…

41. Identifying Even and Odd Functions Algebraically C. Since f(-x) = -f(x), then it’s an ODD function! Graph it…does it appear to be even or odd? First find f(-x)... Then find -f(x)...

42. Determining Where a Function is Increasing, Decreasing, or Constant Look from left to right along a graph and find the parts of the graph that are rising (increasing), falling (decreasing), and horizontal (constant). REMEMBER: these are INTERVALS named by the x coordinates!

43. Example Where is the function decreasing? (-6, -4) U (3, 6) Where is the function increasing? (-4, 0) Where is the function constant? (0, 3)

44. Finding Local Maximums and Local Minimums Maximums are the hills and minimums are the valleys... REMEMBER: these are named by the y coordinates! TRACE to the point, then 2nd – calc – (3) minimum or (4) maximum On your calculator:

45. Example What are the maximums, if any? A max is where y = 2. What are the minimums, if any? There are 2 mins: one at y = 1 and another at y = 0. List the intervals on which f is increasing and decreasing. Decreasing: Increasing: (-1, 1) U (1, 3)

46. Example Use the graphing calculator to graph the function for. Find local maximums and minimums and determine the intervals of increase and decrease. NOTE: Always sketch your graph and fill in the points!

47. Max (-.707, 2.414) Min (.707, -0.414)

48. (-0.707, 2.414) (0.707, -0.414) Increasing: Decreasing: (-2, -0.707) (-0.707, 0.707) (2, 11) (-2, -9) U (0.707, 2)

49. Average Rate of Change of a Function If c is in the domain of a function, f, the average rate of change of f from c to x is defined:, where (In calculus, this expression is called the difference quotient of f at c.) This is just calculating the slope between two points!!

50. Example Find the average rate of change of f(x) = x 2 – 2x + 3 from…...0 to 2:...-1 to 1: When x = -1, y = When x = 1, y = When x = 0, y = When x = 2, y = 6 2 3 3 (-1, 6) and (1, 2) (0, 3) and (2, 3)

51. Graph of a Linear Function y = mx + b It has a constant slope, m The y-intercept is b y and x are the variables, representing vertical and horizontal changes, respectively b = 7 rise run = 3 3 = -3 m = -3 y = -3x + 7

52. Linear Function D: R: Increasing if m > 0 Decreasing if m < 0

53. Constant Function D: R: b m = 0 A special linear function...an even function.

54. Identity Function D: R: m = 1 b = 0 A special linear function where x = y An odd function Increasing from

55. Square (Quadratic) Function D: R: A parabola with vertex at (0, 0)...minimum at 0 An even function Increasing: Decreasing:

56. Cube Function D: R: An odd function Increasing from Intercept at (0, 0) No min or max

57. Square Root Function D: R: Neither even nor odd Increasing from Intercept at (0, 0)

58. Cube Root Function D: R: An odd function Increasing from Intercept at (0, 0) No min or max

59. Reciprocal (Inverse) Function D: R: An odd function No intercepts No min or maxIs discontinuous Decreasing from Has a vertical and horizontal asymptote

60. Absolute Value Function D: R: MATH NUM-1 An even function Intercept at (0, 0) Increasing DecreasingMin at 0

61. Greatest Integer (Step) Function MATH NUM-5 The largest integer less than or equal to x. D: R: integers Neither even nor odd x-intercept at [0, 1) discontinuous

62. Piecewise Functions Functions that are defined by more than one equation and then “pieced” together...

63. Example a. Find the domain of the function. b. Locate any intercepts. c. Graph the function. d. Based on the graph, find the range. (0, 4)

64. Example a. Find the domain of the function. b. Locate any intercepts. c. Graph the function. d. Based on the graph, find the range. (0, -3) (-5/2, 0)

65. Vertical and Horizontal Shifts On the same plane, graph the following functions:

66. Conclusion: A vertical shift occurs whenever a number is added or subtracted to the whole function. It shifts up if a number is added; it shifts down if a number is subtracted. up 2 (x, y + 2) down 2 (x, y - 2) y = f(x) + k y = f(x) - k

67. On the same plane, graph the following functions:

68. right 3 (x + 3, y) left 2 (x - 2, y) NOTE: Always the OPPOSITE effect with x’s!!! Conclusion: A horizontal shift occurs whenever a number is added or subtracted from the x part of a function. It shifts left if a number is added; it shifts right if a number is subtracted. y = f(x + h) y = f(x – h)

69. Compressions and Stretches On the same plane, graph the following functions:

70. narrower (x, 2y) wider (x, ½y) Conclusion: A vertical stretch/compression occurs whenever the whole function is multiplied by a number. A vertical stretch occurs if the number is greater than 1; a vertical shrink occurs if the number is 0<n<1. vertical stretch by factor 2 vertical compression by factor ½ y = af(x)

71. On the same plane, graph the following functions:

72. narrower (1/4x, y) wider (2x, y) NOTE: Always the OPPOSITE effect with x’s!!! Multiply by the RECIPROCAL! A horizontal compression is like a vertical stretch and a horizontal stretch is like a vertical compression... horizontal compression by factor 1/4 horizontal stretch by factor 2

73. Reflections On the same plane, graph the following functions: Conclusion: A reflection on the x-axis occurs when the whole function is multiplied by -1. flipped (x, -y) y = - f(x)

74. On the same plane, graph the following functions: Conclusion: A reflection on the y-axis occurs when the x part of a function is multiplied by -1. flipped (-x, y) y = f(- x)

75. Summary of Graphing Techniques (x, y + k) (x, y - k) (x + h, y) (x - h, y) (x, ay) (1/ax, y) (x, -y) (-x, y)

76. Example Sketch a graph of the function f(x) = (x + 3) 2 – 5 and describe the transformations applied to the parent function to obtain f. f(x) = x 2 f(x) = (x + 3) 2 f(x) = (x + 3) 2 - 5 Shift left 3 units Shift down 5 units

77. Example Find the function that is graphed after the following transformations are applied to f(x) = |x|... 1) shift left 2 units 2) shift up 3 units 3) reflect on y-axis |x + 2| |x + 2| + 3 |-x + 2| + 3 f(x) = |-x + 2| + 3 (-x + 2, y + 3) Check ordered pairs... (-1, 1) (0, 0) (1, 1) (-3, 1) (-2, 0) (-1, 1) (-3, 4) (-2, 3) (-1, 4) (3, 4) (2, 3) (1, 4) subtract 2 from x add 3 to y opposite of x

78. example Graph the function and describe the transformations that were applied to the parent function of f: Parent function: Transformations: shift right 2 shift up 1 vertical stretch by factor 3 (x + 2, 3y + 1) Check ordered pairs...

79. Example Graph the function and describe the transformations that were applied to the parent function f: Parent function: Transformations: shift up 2 reflect over y-axis (-x + 1, y + 2) shift right 1 Check ordered pairs...