Pre-Calculus Mr. Davis

Transcription

1 Pre-Calculus Mr. Davis

2 How to use a Graphing Calculator Applications: 1. Graphing functions 2. Analyzing a function 3. Finding zeroes (or roots) 4. Regression analysis programs 5. Storing values 6. Evaluating derivatives and integrals (Calculus)

3 Graph and analyze this function f x = 1 3 x3 + x 2 1 Graph the function. Describe the curve. Where does it cross the x-axis? In other words: Solve for the roots (or zeros) of the function Locate and identify any local maximums and minimums. In other words: Where does the graph appear to top or bottom out Observe the table of values.

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5 Chapter 1 Functions and Their Graphs Functions 1.3 Graphs of Functions 1.4 Shifting, Reflecting, and Stretching Graphs 1.5 Combinations of Functions 1.6 Inverse Functions 1.7 Exploring Data: Linear Models and Scatter Plots

6 1.2 - Functions What you ll learn: (all on page 16 in textbook) #1 - Decide whether relations between two variables represent a function. #2 - Use function notation and evaluate functions. #3 - Find the domains of functions. #4 - Use functions to model and solve real-life problems #5 - Evaluate difference quotients.

7 Definition of a function A function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain (or input) of the function f, and the set B contains the range (or output). In other words: The function is the rule that is applied to an input and the result is the output. The input is the x variable and the output is the y variable, or f(x).

8 Example of a function The inputs are provided. Please find each of the corresponding outputs. The x values are considered the domain. The y values are considered the range.

9 How can you tell if the relation is a function? Each individual input has to have it s own output. In the chart you will notice that the input of 3 is shown twice. The outputs are different. This is NOT A FUNCTION!

10 Vertical Line Test In order for an expression to be a function, it must pass the vertical line test. The graph may only touch the vertical line once.

11 Explicit vs Implicit Equations Sometimes a function may be written in Implicit Form. (x and y variables are mixed up) Example: x 2 + y = 5 Solve for y to put the function in Explicit Form. Example: y = 5 x 2 Functions are easier to analyze in Explicit Form!

12 Learning Target #1 How do you decide whether relations between two variables represent a function? Check for any disparity between two inputs of the same value Vertical line test

13 Function Notation Function notation allows us to evaluate functions in a more organized way. f x = x 2 + 2x 1 f = f =

14 Evaluating a Function Let h x = x Find each of the following. 1. h 1 2. h( 2) 3. h(r) 4. h( x) 5. h(x) 6. h(x + 2)

15 Piecewise-Defined Functions They are pieces of functions put together to form one complete function. Think of it as a Frankenstein function. For example:

16 Piecewise-Defined Functions Example: f(x) ቊ x2 + 1, x < 0 2x + 1, x > 0

17 Evaluating Piecewise-Defined Functions Find each of the following for the function. x + 3, x < 1 g(x) ቐx 2, 1 x < 3 4, x 3 1. g 0 2. g(1) 3. g(2) 4. g(3)

18 The Domain of a Function The domain of a function is the set of inputs (or the x variable). In some cases, the inputs may be limited.

19 Finding the Domain of a Function Find the domain of each function. 1. f: 3,0, 1,4, 2,2, (4, 1) 2. g x = 3x 2 + 4x h x = 1 x+4 4.Volume of a sphere: V = 4 3 πr3 5. k x = 4 3x

20 Learning Target #3 How do you find the domains of functions? Analyze the possible inputs.

21 Finding the range of a function The range is the set of all possible outputs of a function. Example: (try graphing!) Find the domain and range of the following function: f x = 4 x 2

22 Modeling Real-World Applications Example 7 on page 22 Cell Phone Subscribers 10.75t 20.1, 5 t 7 N(t) ቊ 20.11t 92.8, 8 t 11 5 representing 1995 Use this piecewise-defined function to approximate the number of cell phone subscribers for each of the following years:

23 The Path of a Baseball Example 8 on page 22 Jason Kipnis hit a baseball at a point 3 feet above the ground at a velocity of 100 feet per second and an angle of 45 o. The path of the baseball is given by the function f x = x 2 + x + 3 where y and x are measured in feet. Will the baseball clear a 10-foot fence located 300 feet from home plate?

24 Learning Target #4 How will you use functions to model and solve real-life problems? Understand what the function represents. Use the function to solve for solution.

25 Difference Quotient This is a very important quotient that is used often in calculus. f x+h f x h, h 0 For f x = x 2 4x + 7, find f x+h f(x) h.

26 Homework Page 24 #5,6,27,41,49-61 ODD,70 part c only,95 #83-85 *difference quotients

27 1.3 Graphs of Functions What You ll learn: #6 - Find the domains and ranges of functions. #7 Use the vertical line test. #8 Determine the interval on which functions are increasing, decreasing, or constant. #9 Determine relative max and min values of functions. #10 Identify and graph step functions and other piecewise-defined functions. #11 Identify even and odd functions.

28 Find the domain and range of the function. Then find the values of f 1 & f 2.

29 Find the domain and range of the following function. f x = x 3 Analytically? Graphically?

30 Increasing and Decreasing Functions Functions have intervals in which they many be increasing, decreasing, or neither (constant). For the graph above, determine the intervals on which the function is increasing, decreasing, or constant.

31 Relative Minimum and Maximum Values The x coordinates are used to describe WHERE a min or max value may occur. The y coordinates are used to describe WHAT a min or max value is.

32 Graphing a Piecewise-Defined Function Sketch the graph of f x = ቊ 2x + 3, x 1 x + 4, x > 1

33 Even and Odd Functions A function f is even if for every x in the domain of f, f x = f(x). A function f is odd if for every x in the domain of f, f x = f(x). A function f is considered neither if none of the above occur. Even and odd functions also have graphical symmetry! EVEN ODD NEITHER Symmetry about y-axis Symmetry about origin, or 180 o rotational.

34 Example Even, odd, or neither functions For each function below, indicate if it is an even function, an odd function, or neither even or odd. 1. f x = x f x = 2x 3 + x 3. g x = x 4. h x = x 2 + x 3

35 Homework Page 38 #1-5,17-31 ODD,(35 & 37 part b only),41,48,49-51,87

36 1.4 Shifting, Reflecting, and Stretching Graphs What you ll learn: #12 Recognize graphs of common functions. Library of Functions #13 - Use the vertical and horizontal shifts and reflections to graph functions. #14 - Use nonrigid transformations to graph functions.

37 Family of Functions

38 Graphing Functions Vertical and Horizontal Shifts A graph can be translated up or down with the following action: f x = x 2 now moved up 2 units f x = x A graph can be translated left or right with the following action: g x = x 2 now moved left 3 units g x = x + 3 2

39 Sketching a Graph with Translations Sketch each function with the given translation. 1. f x = x g x = x h x = x

40 Reflecting Graphs This is the second most common transformation for graphs of functions. Notice the red graph is open upwards, while the blue graph is open downwards. The difference between the two graphs is the negative! A negative placed in front of the function reflects the graph about the x-axis! f x f(x)

41 Reflecting Graphs If the variable inside the function is negated, then the function s graph is reflected about the y-axis! f(x) f( x) The red graph is the function e x. The blue graph is the function e x.

42 Exploration Compare the two graphs of the functions: f x = x 2 g x = x 2 Are they the same? If not, how are they different?

43 Stretching or Shrinking Graphs You can have a horizontal stretch or shrink with the following actions: f x = x 2 with a stretch factor of 5 f x = 1 5 x2 f x = x 2 with a shrink factor of 5 f x = 5x 2

44 Examples Graph the function f x = x given each transformation. 1. f x = x 2. f x = x f x = 1 2 x

45 Homework Page 48 #3,4,11,19-26,34,40

46 1.5 Combinations of Functions What You ll Learn: #15 Add, subtract, multiply, and divide functions. #16 Find compositions of one function with another function. #17 Use combinations of functions to model and solve real-life problems.

47 Arithmetic Combinations of Functions No surprises here. It is the same as you d expect! Example: Let f x = 2x 3 and g x = x 2 1. Find each of the following: 1. f x + g(x) 2. f x g(x) 3. f(x) g(x) 4. f(x) g(x)

48 Sum, Difference, Product, and Quotient These are different ways to display the same thing. Sum: f + g Difference: f g Product: fg Quotient: f g x = f x + g(x) x = f x g(x) x = f(x) g(x) x = f(x), assuming that g(x) 0 g(x)

49 Example Given f x = 2x + 1 and g x = x 2 + 2x 1, find (f + g)(x). Then evaluate the sum when x = 2. Solve algebraically (analytically).

50 Graphical Solution Given f x = 2x + 1 and g x = x 2 + 2x 1, find (f g)(x). Then evaluate the difference when x = 2. On your graphing calculator: y 1 = 2x + 1 y 2 = x 2 + 2x 1 y 3 = y 1 y 2

51 Example Product of Two Functions Given f x = x 2 and g x = x 3, find (fg)(x). Then evaluate the product when x = 4.

52 Example Quotient of Two Functions Find ( f g )(x) and (g f )(x) for the functions given by f x = x and g x = 4 x 2. Then find the domains of each.

53 Compositions of Functions We are plugging one function in to the other. Example: Given f x = x 2 and g x = x 3, find: 1. f(g x ) 2. g(f x ) Evaluate (fog)(1). What if x = 0, 2, 3, 4, or 5?

54 Domain of Composite Functions 3 Easy Steps 1. Determine the domain of both functions. 2. Restrict the range of the inner function to fit. (domain of outer function) 3. Modify the domain of the inner function to accommodate the restriction.

55 Finding Domain of Composite Functions Find the domain of the composition (fog)(x) for the functions given by: f x = x 2 9 and g x = 9 x 2 You have to consider the possible outputs of the inner function to determine the domain of the outer function. In other words, the domain of the inner function has to fit the domain of the outer function!

56 (fog)(x) = (gof)(x)? Given f x = 2x + 3 and g x = 1 2 find each composition. A. (fog)(x) (x 3), B. (gof)(x)

57 Try it yourself Find the domain of the composition (fog)(x) for the functions given by: f x = (x + 2) 1 2 and g x = x 2 Remember, the domain of the inner function has to fit the domain of the outer function!

58 Try it yourself Find the domain of the composition (fog)(x) for the functions given by: f x = x and g x = x

59 Identifying a Composite Function Write the function h x = 3x 5 3 as a composition of two functions.

60 Identifying a Composite Function Write the function h x = functions. 1 x 2 2 as a composition of two Any alternative solutions?

61 Try it yourself Write the following functions as a composition of two functions. h x = x 2 + 2x 4 g x = 1 x+5 3

62 Application Bacteria in food Example 11 on Page 57 The number N of bacteria in a refrigerated food is given by: N t = 20T 2 80T + 500, 2 T 14 where T is the temperature of the food in degrees Celsius. When the food is removed from the fridge, the temperature of the food is given by: T t = 4t + 2, 0 t 3 where t is the time (in hours). a) Find the composition N(T t ) and interpret its meaning in context. b) Find the number of bacteria in the food when t = 2 hours. c) Find the time when the bacterial count reaches 2000.

63 Homework Page 58 #5,8,13,36,39,40,45,51,55-62,64-66,79

64 From homework #39 - f x = x + 4 and g x = x 2 A: all real #s #40 - f x = 3 x + 1 and g x = x 3 1 A: all real #s #64 - f x = x + 3 and g x = x 2 A: x 6 #65 - f x = x and g x = x A: x 0 #66 - f x = x 1 4 and g x = x 4 A: all real #s

65 1.6 Inverse Functions What You ll Learn: #18 - Find inverse functions informally and verify that two functions are inverse functions of each other. #19 - Use graphs of functions to decide whether functions have inverse functions. #20 - Determine if functions are one-to-one. #21 - Find inverse functions algebraically.

66 Notation and Purpose If f(x) is the original function, then f 1 (x) would be the inverse. An inverse function is the OPPOSITE of the original function. In other words: it un-does what the original function does.

67 Simple Example f x = x + 4 f 1 x = x 4 Here are some input-output values for the function: (x,y) coordinates 1, 5, 2, 6, (3, 7). Now here are some input-output values for the inverse function: 5, 1, 6, 2, (7, 3)

68 Finding Inverse Functions Informally find the inverse function of f x = 4x. Then verify that both f(f 1 x ) and f 1 (f x ) are equal to the identify function.

69 Graph of Inverse Functions Functions that are inverses of each other will exhibit a reflection about the y=x diagonal line. The composite of the two functions (either way) is the identify function! f x = x!

70 Three Ways to Verify an Inverse 1.Algebraically 2.Graphically 3.Numerically

71 Try this Show that the functions are inverse functions of each other algebraically, graphically, and numerically. f x = 2x 3 1 and h x = 3 x+1 2

72 One-to-one Function

73 Testing for an Existence of Inverses Because inverses have to be functions too, we need a quick test to determine the existence of inverses. Functions need to be one-to-one for its inverse to exist as a function. Horizontal Line Test

74 Finding an Inverse Function Algebraically Three Steps 1. Switch x and y 2. Solve for y 3. Verify Meaning f f 1 x = f 1 f x = x In other words, you simplify the composition of the two functions (BOTH WAYS) down to x.

75 Finding an Inverse Algebraically Find the inverse function of f x = 5 3x 2.

76 Finding an Inverse Algebraically Find the inverse function of g x = x 3 4.

77 Finding an Inverse Algebraically Find the inverse function of h x = 2x 3.

78 Homework Page 69 #7,9-11,15,16,21-24,29-33,53,61

79 1.7 Exploring Data: Linear Models and Scatter Plots What You ll Learn: #22 - Construct scatter plots and interpret correlation. #23 - Use scatter plots and a graphing utility to find linear models for data. *Worksheet Handout* HW Page 77 #1,3-6 a&b,7,13

80 Chapter 1 Review Page 83 #17,35,37,41,47,49,51,52,57,58,60,61,65, 67,68,70,71,72,85,89,92,93,98, point HW assignment