1. CHAPTER 1 FUNCTIONS AND GRAPHS

2. Quick Talk What can you tell me about functions based on your peers’ lessons? What can you tell me about graphs based on your peers’ lessons?

3. Review about factoring

4. Challenge: Could you factor these?

5. Quick Talk: What does factoring allow you to find?

6. Potential Answers Rational Zeros when y=0

7. A mathematical model is a mathematical structure that approximates phenomena for the purpose of studying or predicting their behavior. There are 3 types of model: Numerical Models Algebraic model Graphical models

8. The most basic type: Numerical Model Numerical Model: use of numbers or data are analyzed to gain insights into phenomena

9. Group work: Numerical Model Example #1 YearTotalMaleFemale 1980456324132 1985480328152 1990532360172 1995690510180 2000719520199 The chart shows growth in the number of engineers from 1980-2000. Is the proportion of female engineers over the years increasing? Explain

10. Answer You have to be careful with the last example, the word “proportion” means what? Once we converted all the data into ratios, we see that there is an increase from 1980-1995, but there is a drop from 1990- 1995, then from 1995-2000 it increased again. From the data, the peak is at 1990, 32.3% YearTotalFemaleActual ratio 1980456132.289 1985480152.317 1990532172.323 1995690180.261 2000719199.277

11. Algebraic Model Algebraic model uses formulas or equations to relate variable quantities associated with the phenomena being studied.

12. Group Work: Algebraic Model Example #1 Note: Think about what are you comparing. What “formula” can you use? Or do you have to create one?

13. Answer

14. Group Work: Algebraic Model Example #2 You went to Gamestop, it just happens every game you buy is discounted 15%( ) off the marked price. The discount is taken at the sales counter, and then a state sales tax of 7.5% and local tax of 1% is added on. Questions: 1) If you have $20, could you buy a game marked at $24.99? 2) If you are determined to spend no more than $175, what’s the maximum total value of your marked purchases be?

15. Answer 1) Identify your variables (ex: let m=market price, d=discounted price, k=constant, t=taxes, s=total sale price) Your general equation should be: d=km s=d+td When you substitute: s=km+t(km) Now determine the values for each variable: k=.85 m=? d=? t=.085 s=? Your actual equation (with numbers) should be: d=.85m s=.85m+.085(m) 1) m=24.99, s=.85(24.99)+.085(24.99), s=21.24+2.12, s= 23.36 Since you only have $20, 20 < 23.36, therefore, you can not buy the game 2) s=175, so 175=.85(m)+.085(m), m=187.17, so the maximum total value of the market price should be at most $187.17

16. Graphing Model Graphing model is a visible representation of a numerical model or an algebraic model that gives insight into the relationships between variable quantities.

17. Graphical model Example #1 Graph this. What does it look like? Can you find an algebraic model that fits?

18. Answer

19. Graphical Model example #2 Time (t)05101520 Females (f)3.84.45.55.96.7 Create a graphic model, that fits the best with these data sets.

20. Answer

21. Group work Situation:

22. Answer

23. Group Work Solve for this equation

24. Answer X=0 or x=5/2 or x=-2/3

25. Group Work Solve this

27. Homework Practice Pgs 81-84 #1-9odd, 11-18, 29, 31, 35,43, 45, 48, 50

28. BASIC PARENT FUNCTIONS

29. Quick Talk: What makes a function? What does it consists of?

30. Answer One to one / passes vertical line test Y and x Equation Domain Range Note: y is aka f(x)

31. Quick Talk: From what you just learned, which of these is not a function?

32. Quick Talk: What are parent functions?

33. Parent Function is the simplest function with characteristics. It is without any “transformations” or “shifting”

34. 12 parent functions: As you are graphing it, please draw an arrow at the end.

35. The Identity Function Slide 1- 35

36. The Squaring Function Slide 1- 36

37. The Cubing Function Slide 1- 37

38. The Reciprocal Function Slide 1- 38

39. The Square Root Function Slide 1- 39

40. The Exponential Function Slide 1- 40

41. The Natural Logarithm Function Slide 1- 41

42. The Sine Function Slide 1- 42

43. The Cosine Function Slide 1- 43

44. The Absolute Value Function Slide 1- 44

45. The Greatest Integer Function Slide 1- 45

46. The Logistic Function Slide 1- 46

47. Important!!! Read this!!! I neeeeeeedddddddd you guys to recognize, understand and know the parent functions!!!!! Why? Because it will make finding domain and range easier.

48. What “shifts” can you possibly have on a parent function? You can shift a function left or right (horizontal shifts) You can shift a function up or down (vertical shifts) Function can be steeper or flatter (linear) Function can be wider or narrower (horizontal shrink/vertical stretch or horizontal stretch/vertical shrink) Function can flip across x or y axis

49. Group Activity: graphing! Each group will select a member to put it on the whiteboard. With t chart. X:[-5,5]

50. Shifts review

51. Name the parent function then the shifts

52. Answer Shift right 5 Shift down 9 Vertical stretch of 3/2 or horizontal shrink of 2/3

53. Homework Practice P147 #1,4,5,7,9,17,22

54. Domain and Range Domain are all the input values of the function. Range are all the output of the function.

55. Looking at Domain and Range Graphically What is the parent function? What is the domain? What is the range?

56. Answer

57. Group Work: What is the domain and range?

58. Answer Domain: [-5,5] Range: [-3,3]

59. Group Activity: Find the Domain and Range of the 12 parent functions. Note: The answers are not on my slides, I will be showing my answers right now in class. If you miss this, get it from me or another person from your class.

60. As you can see from the step function, not every domain or range are continuous.

61. Slide 1- 61 Example Identifying Points of Discontinuity Which of the following figures shows functions that are discontinuous at x = 2?

62. Slide 1- 62 Continuity

63. Finding Domain and range Algebraically

64. How do you find the domain algebraically?

65. Group Work: Determining the domain and range

67. Look at our example #1 and #3 We have something unique. It is called an asymptote.

68. Slide 1- 68 Horizontal and Vertical Asymptotes

69. Asymptotes Remember the graph will get infinitely close to the asymptotes but will NEVER intersect with it.

70. Slide 1- 70 Increasing, Decreasing, and Constant Function on an Interval A function f is increasing on an interval if, for any two points in the interval, a positive change in x results in a positive change in f(x). A function f is decreasing on an interval if, for any two points in the interval, a positive change in x results in a negative change in f(x). A function f is constant on an interval if, for any two points in the interval, a positive change in x results in a zero change in f(x).

71. Slide 1- 71 Increasing and Decreasing Functions

72. Group work: Find the Increase and Decrease

73. Shown in class: How do you graph it?

74. Slide 1- 74 Answer:

75. Slide 1- 75 Local and Absolute Extrema A local maximum of a function f is a value f(c) that is greater than or equal to all range values of f on some open interval containing c. If f(c) is greater than or equal to all range values of f, then f(c) is the maximum (or absolute maximum) value of f. A local minimum of a function f is a value f(c) that is less than or equal to all range values of f on some open interval containing c. If f(c) is less than or equal to all range values of f, then f(c) is the minimum (or absolute minimum) value of f. Local extrema are also called relative extrema.

76. In another word You may have multiple relative/local maximum and relative/local minimum. They are located where the slopes are = 0 (important concept in calculus) Absolute “maximum” or “minimum” is where it is the most “top” or “bottom point” of the function.

77. Slide 1- 77 Lower Bound, Upper Bound and Bounded A function f is bounded below of there is some number b that is less than or equal to every number in the range of f. Any such number b is called a lower bound of f. A function f is bounded above of there is some number B that is greater than or equal to every number in the range of f. Any such number B is called a upper bound of f. A function f is bounded if it is bounded both above and below.

78. In another word, Function is bounded below, if there is an “absolute minimum” or if there is a “floor” Function is bounded above, if there is an “absolute maximum” or if there is a “ceiling”

79. End Behavior Asymptote

80. Example: Find end behavior asymptote

81. Answer:

82. Group work: Find end behavior asymptote

83. Answer:

84. Symmetry Symmetry: a line where you can fold one side onto the other.

85. Slide 1- 85 Symmetry with respect to the y-axis

86. Slide 1- 86 Symmetry with respect to the x-axis

87. Slide 1- 87 Symmetry with respect to the origin

88. Slide 1- 88 Example Checking Functions for Symmetry

89. Slide 1- 89 Example Checking Functions for Symmetry

90. Group work: Look at the 12 parent functions. Find the boundedness, location of local/absolute min/max, where it is increasing, decreasing or constant.

91. Group work:

92. Answer shown in class

93. Homework Practice

94. BUILDING FUNCTIONS FROM FUNCTIONS

95. Overview It is important how you can put functions together.

96. Slide 1- 96 Sum, Difference, Product, and Quotient

97. Slide 1- 97 Example Defining New Functions Algebraically

98. Group work: From the last examples, find the following: Domain: Range: Continuous: Increase/decrease: Symmetric: Boundedness: Max/min: Asymptotes: End behavior:

99. Slide 1- 99 Composition of Functions

100. Note: Composition is like “function within a function”

101. Composition Examples

102. Answer:

103. Group Work:

104. Answer shown in class

105. Group Work:

106. Answer

107. Working backward

108. Possible Answers

109. Group Work:

110. Answer

111. Word problem A high-altitude spherical weather balloon expands as it rises due to the drop in atmospheric pressure. Suppose that the radius r increases at the rate of 0.03 inches per second and that r=48 inches at time t=0. Determine an equation that models the volume V of the balloon at time t and find the volume when t=300 seconds.

112. Answer

113. Word Problem #2 A satellite camera takes a rectangular shaped picture. The smallest region that can be photographed is a 5km by 7km rectangle. As the camera zooms out, the length l and width w of the rectangle increase at a rate of 2 km/s. How long does it take for the area A to be at least 5 times its original size?

114. Answer

115. Implicitly: Something not directly expressed

117. Group Work: (try factor it!)

118. Answer

119. Homework Practice P124 #2,4,6,7,12,16,19,23,24,34,37,44

120. PARAMETRIC RELATIONS AND INVERSES

121. Parametric Equation A function that define both elements of the ordered pair (x,y) in terms of another variable t.

122. Example of Parametric

123. Answer

124. Group Work: Graph Parametric Equation

125. Answer t(x,y) -3(3,-2) -2(0,-1) (-1,0) 0(0,1) 1(3,2) 2(8,3) 3(15,4)

126. How to do parametric equation with the calculator.

127. Slide 1- 127 Inverse Relation The ordered pair (a,b) is in a relation if and only if the pair (b,a) is in the inverse relation.

128. Slide 1- 128 Horizontal Line Test The inverse of a relation is a function if and only if each horizontal line intersects the graph of the original relation in at most one point.

129. Slide 1- 129 Inverse Function

130. How do you find the inverse?

131. Group work: Find the inverse

132. Slide 1- 132 Group Work:

133. Slide 1- 133 Answer

134. Slide 1- 134 The Inverse Reflection Principle The points (a,b) and (b,a) in the coordinate plane are symmetric with respect to the line y=x. The points (a,b) and (b,a) are reflections of each other across the line y=x.

135. Slide 1- 135 The Inverse Composition Rule

136. Group Work

137. Homework Practice P135 #2,5,9-12, 14,16,22,27,29,33

138. MODELING WITH FUNCTIONS Please bring your book

139. Quick Talk: Name the following formulas 1) Volume of Sphere 2) Volume of Cone 3) Volume of Cylinder 4) Surface Area of Sphere 5) Surface Area of Cone 6) Surface Area of Cylinder 7) Area of circle 8) Circumference

140. Answer

141. Slide 1- 141 Example A Maximum Value Problem

142. Answer

143. Group Work Grain is leaking through a hole in a storage bin at a constant rate of 5 cubic inches per minute. The grain forms a cone-shaped pile on the ground below. As it grows, the height of the cone always remains equal to its radius. If the cone is one foot tall now, how tall will it be in one hour?

144. Answer

145. I hope you bring your book Classwork P160 #1-35 every other odd