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3 Analyzing Functions Raja Almukahhal Larame Spence Mara Landers Nick Fiori Art Fortgang Melissa Vigil Say Thanks to the Authors Click (No sign in required)

4 To access a customizable version of this book, as well as other interactive content, visit CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform. AUTHORS Raja Almukahhal Larame Spence Mara Landers Nick Fiori Art Fortgang Melissa Vigil Copyright 2013 CK-12 Foundation, The names CK-12 and CK12 and associated logos and the terms FlexBook and FlexBook Platform (collectively CK-12 Marks ) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License ( licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the CC License ), which is incorporated herein by this reference. Complete terms can be found at Printed: October 1, 2013

5 Chapter 1. Analyzing Functions CHAPTER 1 Analyzing Functions CHAPTER OUTLINE 1.1 Relations and Functions 1.2 Domain and Range 1.3 Intervals and Interval Notation 1.4 Average Rate of Change 1.5 Minimums and Maximums 1.6 Discrete and Continuous Functions 1.7 Increasing and Decreasing Functions 1.8 Limits and Asymptotes 1.9 Infinite and Non-Existent Limits 1.10 Linear and Absolute Value Function Families 1.11 Square and Cube Function Families 1.12 Vertical and Horizontal Transformations 1.13 Stretching and Reflecting Transformations 1.14 Combining Transformations 1.15 Operations on Functions 1.16 Composition of Functions 1.17 Linear and Quadratic Models 1.18 Cubic Models Introduction This section focuses on investigating functions both graphically and mathematically. Lessons in this section specifically teach: Identifying the maximum and minimum values of a function both by investigating a graph and using Algebra. Identifying the rate of change of a function Understanding the concept of limit Recognizing and using asymptotes Understanding and using function notation Parent functions and the use of transformations Recognizing the difference between functions and relations Understanding the concept of intervals and the use of interval notation By the time you finish this section, you should have a fresh understanding of what functions are, and how to manipulate them to find specific bits of information. 1

6 1.1. Relations and Functions Relations and Functions Here you will learn the meaning of the term function, and how to demonstrate the correlation between relationships and functions. Suppose you wanted to predict the cost of going to see a movie at the theater, you text a number of your friends who have seen movies recently to ask how much it cost them, here are the responses: "$14.50 :-(" "$ $3.50 for popcorn" "five bucks - dollar theater" "$17.50 : -( broke now" "$12.75 loved the 3D!" Can you accurately predict the cost of going to a movie from these responses? Why or why not? Watch This MEDIA Click image to the left for more content. Khan Academy - Relations and Functions Guidance Consider two situations shown in the boxes below: TABLE 1.1: Situation 1: You are selling candy bars for a school fundraiser. Each candy bar costs $3.00 Situation 2: You collect data from several students in your class on their ages and their heights: (18,65"), (17,64"), (18,67"), (18,68"), (17,66") In the first situation, let the variable x represent the number of candy bars that you sell, and let y represent the amount of money you make. If you sell x candy bars, you will make y = 3x dollars. For example, if you sell 25 candy bars, you will make 3(25) = $ Notice that you can use the number of candy bars you sell to predict how much money you will make. Now consider the second situation. Can you similarly use the data to predict specific height, based on age? No, this is not the case in the second situation. For example, if a student is 18 years old, there are several heights that the student could be. 2

7 Chapter 1. Analyzing Functions The first situation is an example of a function, and the second example is not a function. TABLE 1.2: A function is a relationship where each input number corresponds to one and only one output number. In the first situation, for each different number of candy bar sales you input, there is one and only one output number representing your profit. In the second situation, if you input "18 years", there are multiple outputs, so you can t identify a specific relationship between age and height. See the difference? It is important to note that both situations above are relations. A relation is simply a relationship between two sets of numbers or data. For example, in the second situation, we created a relationship between students ages and heights, just by writing each student s information as an ordered pair. In the first situation, there is a relationship between the number of candy bars you sell and the amount of money you make. The first example is different from the second because it represents a function: every x is paired with only one y. Functions may be presented in many ways. Some of the most common ways to represent functions include: sets of ordered pairs, equations, and graphs. The figure below shows the same function depicted in three different ways. Example A: Determine if each relation is a function: TABLE 1.3: Representation Set of ordered pairs Equation Graph Example (1,3), (2,6), (3,9), (4,12) (a subset of the ordered pairs for this function) y = 3x Solution: In the first representation above, we are given a set of ordered pairs. To verify that this is a function, we must ensure that each x-value is associated with a single y-value. In this example, the first number in each pair (the x-value) is different, so we can be certain that there are no cases where a particular x is associated with more than one y. In the second representation, the equation of a line, it is apparent that any number put in place of x will result in a different y, since the x number is simply being multiplied by 3. The third representation above is a graph. A good way to determine whether a relation is a function when looking at a graph is by doing a "vertical line test". If a vertical line can be drawn anywhere on the graph such that the line crosses the relation in two places, then the relation is not a function. If all possible vertical lines will only cross the relation in one place, then the relation is a function. This works because if a vertical line crosses a relation in more than one place it means that there must be two y values corresponding to one x value in that relation. For example, the graph above of y = 3x shows it is a function because any vertical line that is drawn only crosses the relation in 3

8 1.1. Relations and Functions one place. Conversely, the graph below of x = y 2 shows it is not a function because a vertical line can be drawn that crosses the relation in two places. Example B: Determine if each relation is a function TABLE 1.4: a) (2, 4), (3, 9), (5, 11), (5, 12) b) Function defined as: Solution: a. (2, 4), (3, 9), (5, 11), (5, 12) This relation is not a function because 5 is paired with 11 and with 12. b. (referring to image) This relation is a function because every x is paired with only one y. A vertical line through the graph will always only encounter a single point. Example C: Remember the question about movie tickets at the beginning of the lesson? Does the data you received from your friends represent a function? Can you use the data to predict the cost of going to a movie yourself? Solution: If we were to organize the information we received into ordered pairs, it might look something like: (1, 14.5)(1, 8.75)(1, 5)(1, 17.5 where each x value represents the number of tickets bought, and each y value represents the price. 4

9 Chapter 1. Analyzing Functions Since there are many different y values for the only x value, it is definitely not a function. It should be clear now that the information received from friends text messages cannot really be used to accurately predict the cost of a movie. Vocabulary A relation is a comparison of two or more sets of values. A function is a relation of two or more sets of values in which each input number corresponds to one and only one output number. Guided Practice Determine if each relation is a function: 1) ( 1,4)(0,3)(1,5)(1,7)(2,15) 2) y = x 3) (2, 0)(4, 1)(2.1, 4)(1, 4)(4, 1) 4) y = 4x 5) x = y Solutions 1) There are two different outputs or y-values for the input or x-value of 1. Because we cannot know whether 1 should go with 5 or 7 at any given time, this relation is not a function. 2) Since y = x, any time a number is chosen to represent x, that, and only that, number becomes y. From this it is apparent that each input has one and only one output: This relation is a function. 3) Don t be fooled! This is a function, there is only one unique output for each input. The fact that both x values 2.1 and 1 are associated with y value 4 does not mean that 2.1 and 1 don t have a specific associated value. Also, not matter how close two x s (2 and 2.1, for instance) may be, if they are not exactly the same, they don t affect the definition of a function. 4) This is a function, very similar to #2. Any value chosen for x has one and only one associated value for y (4 times as big). 5) This is not a function. This graph looks like a "<", with the point on the origin. Any value chosen for x will have 2 associated y values. For instance: 4 = -4 and 4 = 4. Practice 1. What is the definition of a function? 2. Can a function definition be written in the form x = 3y instead of y = 3x? 3. Is it mandatory for a function to have both an input and an output? 4. Can a statement be a function if there is only one input and output? 5. Give an example of a relation that is not a function, and explain why it is not a function. For Questions 6-14, identify each relation as either a function, or not a function: 6. (2, 4) (4, 6) (6, 8) (3, 4) (5, 7) (8, 2) 7. (-1, 6) (0, 4) (-4, 0) (-1, -6) (-3, -8) 5

10 1.1. Relations and Functions (Jim, Kitty) (Joe, Betty) (Brian, Alice) (Jesus, Anissa) (Ken, Kelli) 11. (Jim, Alice) (Joe, Alice) (Brian, Betty) (Jim, Kitty) (Ken, Anissa) 12. 6

11 Chapter 1. Analyzing Functions At a Prom dance, each boy pins a corsage on his date. Is this an example of a function? 16. Later, at the same dance, Cory shows up with two dates, does this change the answer? 7

12 1.2. Domain and Range Domain and Range Here you will learn to identify the independent variable, (the domain) and the dependent variable (the range) of a function. You and your 3 friends are are debating which theater to choose for a trip to the movies on Friday. There are 4 theaters in town, all with different prices for tickets. The most expensive theater charges $12.50 per person, but has stadium seating and a screen 5 stories tall. The cheapest is only $5.50 per ticket, but only plays older movies and is much less comfortable. The other two are between the extremes at $8.50 and $9.75 each. If you were to graph the total costs for the 4 of you based on which theater you choose, which values would represent the domain and which the range? Watch This Embedded Video: MEDIA Click image to the left for more content. James Sousa Determining Domain and Range Guidance Independent variable, domain The domain of a function is defined as the set of all x values for which the function is defined. For example, the domain of the function y = 3x is the set of all real numbers, often written as R. This means that x can be any real number. Other functions have restricted domains. For example, the domain of the function y = x is the set of all real numbers greater than or equal to zero. The domain of this function is restricted in this way because the square root of a negative number is not a real number. Therefore, the domain is restricted to non-negative values of x so that the function values will be defined. The variable x is often referred to as the independent variable, while the variable y is referred to as the dependent variable. We talk about x and y this way because the y values of a function depend on what the x values are. That is why we also say that y is a function of x. For example, the value of y in the function y = 3x depends on what x value we are considering. If x = 4, we can easily determine that y = 3(4) = 12. Dependent Variable, Range The range of a function is defined as the set of all y values for which a function is defined. Just as we did with domain, we can examine a function and determine its range. Again, it is often helpful to think about what restrictions there might be, and what the graph of the function looks like. Consider for example the function y = x 2. The domain of this function is R, all real numbers, but what about the range? 8

13 Chapter 1. Analyzing Functions The range of the function is the set of all real numbers greater than or equal to zero. This is the case because every y value is the square of an x value. If we square positive and negative numbers, the result will always be positive. If x = 0, then y = 0. We can also see the range if we look at a graph of y = x 2. Notice below that the y values are all greater than or equal to zero. Example A State the domain of each function: TABLE 1.5: a. y = x 2 b. 1 x c. (2, 4), (3, 9), (5, 11) Solution: a. y = x 2 The domain of this function is the set of all real numbers. There are no restrictions. b. 1 x The domain of this function is the set of all real numbers except x = 0. The domain is restricted this way because a fraction with denominator zero is undefined. 9

14 1.2. Domain and Range c. (2, 4), (3, 9), (5, 11) The domain of this function is the set of x values: {2, 3, 5}. Example B State the domain and range of the function y = 2 x Solution: The domain and range of the function y = 2 x For this function, we can choose any x value except x = 0, since we cannot divide a number by zero. Therefore the domain of the function is the set of all real numbers except x = 0. The range is also restricted to the non-zero real numbers, but for a different reason. Because the numerator of the fraction is 2, the numerator can never equal zero, so the fraction can never equal zero. Example C Determine the domain of the function y = x, shown in the graph below. Solution First consider what restrictions there might be and then look at the graph. We can see that y = x has a domain of all real numbers greater than or equal to zero because the graph only exists for x values that are positive, since there are no real square roots of negative numbers. TABLE 1.6: Concept question wrap-up If you were to graph the total costs for the 4 of you based on which theater you choose, which values would represent the domain and which the range? One function that describes this situation would be: P(t) = 4t, where P(t) would be total Price based on per-ticket price, and it is equal to 4 x ticket price. In this case, your domain would be: {$5.50, $8.50, $9.75, $12.50} since these are the prices you would input in place of the independent variable, x to get the total price for each theater. The range would be: {$22.00, $34.00, $38.00, $50.00} since these are the output values given by the dependent variable, y. 10

15 Chapter 1. Analyzing Functions Vocabulary The domain of a function is the set of all x values under which the function is defined. The x variable is also called the independent variable. The range of a function is the set of all y values under which the function is defined. The y variable is also called the dependent variable. Guided Practice Questions 1) Determine if each relation is a function: TABLE 1.7: a) (-1,4), (0, 3), (1, 5), (1, 7), (2, 15) b) y = x 2) State the domain and range of each relation in question A. 3) Give an example of a function for which the domain and range are equivalent to each other. 4) Imagine a playground full of children where all of the boys are wearing baseball caps. The function "boys in the playground" could be used to identify kids wearing hats. In this case, which group would be the domain, and which the range? In other words, what group of kids is the input, and what group the output? 5) Would the example from question 4 be a function if used backwards? Which group would be the domain and which the range in that case? Answers 1) a. Not a function, since there are two different y values for the same x value (1). b) Is a function. Since x and y are the same number (equal), any number chosen for x will have one and only one y. 2) a) The domain is the set of x values: {-1, 0, 1, 2}. The range is the set of y values: {3, 4, 5, 7, 15} b) The domain or x value can be any real number. D:R. The range or y value can therefore also be any real number. R: R 3) Answers will vary. y = x is an example. 4) In this example, "boys in the playground" is the domain, since you would use that group to identify hat-wearers. 5) It could be, it depends on whether or not there are girls wearing hats! If there are girls in the playground wearing hats, then inputing "kids wearing hats" would not return only "boys in the playground". In either case, however, "kids wearing hats" would be in domain in this version of the function. Practice Determine the Domain and the Range of the relations. 1. Relation: {(0,4) (3, 20) (90, 33)} 2. Relation: {(3, -4) (6, 37) (10, -10) (-31, 2)} 3. Tina s car travels about 30 miles on one gallon of gas. She has between 10 and 12 gallons of gas in the tank. Find the domain and range of the function to calculate how far she can drive. 4. Joe and three of his friends plan on going bowling and plan on bowling one or two games each. Each game costs $2.75. Find the domain and range of the function calculating the cost of the trip. 11

16 1.2. Domain and Range 5. Bob had a summer job that paid $10.00 per hour and he worked between hours every week. His weekly salary can be modeled by the equation S = 10h, where S is his weekly salary and h is the number of hours he worked per week. What is the independent variable for this problem? Describe the domain and range for this problem. 6. What does each value in the ordered pair (20, 200) mean in context of the previous problem? 7. Which group of students represents the domain, and which the range, in these ordered pairs? (Jim, Kitty) (Joe, Betty) (Brian, Alice) (Jesus, Anissa) (Ken, Kelli) State the domain and range: 8. y = x 9. x = y 10. y = x x = y

17 Chapter 1. Analyzing Functions

18 1.3. Intervals and Interval Notation Intervals and Interval Notation Learn to identify real functions, and recognize closed and open intervals. Learn to interpret and express intervals in interval notation. Suppose you and 2 of your friends were out for lunch and decide to buy tacos. Together you have $15 to spend on lunch, and tacos are $1.25 each. It is clear that the total cost could be graphed as a function of the number of tacos purchased, but how would you specify that the graph should not include values greater than $15 or less than $3.75 (one taco each)? Watch This Embedded Video: MEDIA Click image to the left for more content. James Sousa Interval Notation Guidance Real Values and Intervals A function is defined as a real function if both the domain and the range are sets of real numbers. Many of the functions you have likely encountered before are real functions, and many of these functions have Domain = R. Consider, for example, the function y = 3x. A section of the graph of this function is shown below. 14

19 Chapter 1. Analyzing Functions You may already be familiar with the graphs of lines. In particular, you may already be in the habit of placing arrows at the ends. We do this in order to indicate that the line will continue forever in both the positive and negative directions, both in terms of the domain and the range. The line above, however, only shows the function y = 3x on the interval [-3, 3]. The square brackets indicate that the graph includes the endpoints of the interval, where x = -3 and x = 3. We call this a closed interval. A closed interval contains its endpoints. In contrast, an open interval does not contain its endpoints. We indicate an open interval with parentheses. For example, (-3, 3) indicates the set of numbers between -3 and 3, not including -3 and 3. You may have noticed that the open interval notation looks like the notation for a point (x, y) in the plane. It is important to read an example or a homework problem carefully to avoid confusing a point with an interval! The difference is generally quite clear from the context. The table below summarizes the kinds of intervals you may need to consider while studying functions and their domains: TABLE 1.8: Interval notation Inequality notation Description [a,b] a x b The value of x is between a and b, including a and b, where a, b are real numbers. (a,b) a < x < b The value of x is between a and b, not including a and b. [a,b) a x < b The value of x is between a and b, including a, but not including b. (a,b] a < x b The value of x is between a and b, including b, but not including a. (a, ) x > a The value of x is strictly greater than a. [a, ) x a The value of x is greater than or equal to a (,a) x < a The value of x is strictly less than a (,a] x a The value of x is less than or equal to a. 15

20 1.3. Intervals and Interval Notation Example A Identify the sets described: a.) ( 3,9] b.) [ 23,12] c.) (,0) Solution a.) The set of numbers between -3 and 9, not including the actual value of -3, but including 9. b.) The set of numbers between -23 and 12, including the values -23 and 12. c.) All numbers less than 0, not including 0 itself. Example B Sketch the graph of the function f (x) = 1 2x 6 on the interval [-4, 12). Solution The figure below shows a graph of f (x) = 1 2x 6 on the given interval: Example C Describe the specified intervals, use interval notation: a.) All positive numbers b.) The numbers between negative eight and two hundred forty two, including both c.) All negative numbers, zero, and the positive numbers up to and including nine. Solution a.) (0,+ ) Zero is neither positive nor negative, so the ( is used to specify that zero is not included. Since there is no maximum positive number, we specify that infinity is the upper value, and use ) since it cannot be reached. b.) [ 8,242] 16

21 Chapter 1. Analyzing Functions The [ is used on both ends, since both values are included. c.) (,9] The ( denotes that negative infinity cannot be reached, and ] on the other end specifies that 9 is included in the set. TABLE 1.9: Concept question wrap-up To specify that the graph of the cost of lunch only includes values between $3.75 and $15.00, specify the interval of the domain as: [3.75, 15]. Vocabulary An interval is a specific part of a function, which may be evaluated separately from the rest of the function. A closed interval includes the endpoints of the interval. An open interval does not include the endpoints of the interval. A real function is a function where both the domain and range are the set of all real numbers. Guided Practice Questions 1) Describe the set shown in the image using interval notation 2) Describe the specified intervals, use interval notation: a) All negative numbers b) The numbers between five and twelve, including five, but not twelve. c) Negative numbers down to negative six, zero, and all positive numbers. 3) Describe the domain in the sets in the images using interval notation: TABLE 1.10: a) b) 17

22 1.3. Intervals and Interval Notation 4) Describe the range in the sets in the images above using interval notation. Solutions 1) (,3)(0, ) The set is opened with "(", since neg infinity cannot be reached, then closed with ")", since 3 is not included. The set is re-opened with "(" since 0 is not included, and finally closed with ")" since pos infinity cannot be reached either. 2) a) (,0) Zero is neither positive nor negative, so the ) is used to specify that zero is not included. Since there is no maximum negative number, we specify that infinity is the lower value, and use ( since it cannot be reached. b) [5,12) The [ is used to open the set, since 5 is included, but ")" is used to close, since 12 is not. c) (,9] The ( denotes that negative infinity cannot be reached, and ] on the other end specifies that 9 is included in the set. 3) a) The domain is the set of x values starting with the included -6 and ending at 4, which is not included: [-6, 4) b) As above: [-6, 7) 4) a) The range is the set of y values from -3 (not included) to 4 (included): (-3, 4] b) As above: [-1, 6) Practice Write the following in interval notation x < <x <2 3. x >-3 4. x 2 Solve and put your answer in interval notation x + 3 <1 6. 7x + 4 2x - 6 For each number line, write the given set of numbers in interval notation

23 Chapter 1. Analyzing Functions Name the domain and range for each relation using interval notation

24 1.3. Intervals and Interval Notation Express the following sets using interval notation, then sketch them on a number line 13. {x : 1 x 3} 14. {x : 2 x <1} 15. A is the set of all numbers bigger than 2 but less than or equal to {x: 3 <x < } 20

25 Chapter 1. Analyzing Functions 1.4 Average Rate of Change Here you will learn about identifying the average slope for a portion of a graph, called an interval. Suppose you and your family are planning to drive to DisneyWorld. You know that the trip includes the following approximate sections: 200 Miles East 100 Miles South 350 Miles East 50 Miles North 150 Miles East 450 Miles South How could you generalize the description of the journey? Watch This Embedded Video: MEDIA Click image to the left for more content. - James Sousa: Average Rate of Change Guidance Average Rate of Change Consider the following situation: you are on a week long road trip with your friend. When you begin to drive on the second day, you have already driven a total of 200 miles. After 6 hours of driving on the second day, you have driven a total of 500 miles. On average, how many miles did you drive per hour on the second day of the trip? The graph below shows this situation, with the x axis representing the number of hours driving (on the second day), and the y axis representing the number of miles driven. The first point on the graph, (0, 200), says that at the beginning of the second day you have already driven 200 miles. The second point on the graph, (6, 500), says that after 6 hours of driving on the second day you have driven 500 miles total. 21

26 1.4. Average Rate of Change Notice that in total, during your 6 hours of driving, you have driven 300 miles. The rate at which you drove is 300 miles in 6 hours, or 50 miles per hour. We refer to this rate as the average rate of change because it is an average across the 6 hours. That is, you did not necessarily drive 50 miles every hour. There could have been one hour where you drove 70 miles and another hour where you drove only 30 miles. We can represent the average rate of change on the graph by indicating how much each quantity has changed: The y values increased by 300, and the x values increased by 6. The average rate of change is the ratio of these changes in each variable. This is how we can define average rate of change in general: Average rate of change = change in y change in x We can examine the average rate of change of a function, whether it is represented as data, as in the previous example, or by an equation. Notice that the average rate of change of the function f (x) = 4x is the slope of the line, 4. While a linear function has a constant slope, other functions, such as f (x) = x 2, will not. You will explore this idea in greater detail in your study of calculus. Example A: Find the average rate of change on the given interval: f (x) = x 2 on [0, 2] Solution: f (x) = x 2 The endpoints of the interval are (0, 0) and (2, 4). Therefore the change in y is 4 and the change in x is 2. The average rate of change is 4/2 = 2. Example B: Find the average rate of change on the given interval: f (x) = 4x on [1, 7] Solution: f (x) = 4x The endpoints of the interval are (1, 4) and (7, 28). Therefore the change in y is 28-4 = 24 and the change in x is 7-1 = 6. The average rate of change is 24/6 = 4. 22

27 Chapter 1. Analyzing Functions Example C: What is the average rate of change of the function f (x) = 3x 2 on the interval [2, 5]? Solution: The two points are (2, 12) and (5, 75). The average rate of change is 63/3 = 21. TABLE 1.11: Concept question wrap-up One way to generalize the information about your trip is to identify your average rate of change in position from home to DisneyWorld. Average Rate of Change = (Change in y) / (Change in x) The trip segments were: 200 Miles East 100 Miles South 350 Miles East 50 Miles North 150 Miles East 450 Miles South If we assume that travelling South is -y, and traveling East is +x: The total change in y = = -500 The total change in x = = 700 Your trip includes a total of apx 1200 miles of travel, and you will average 5 miles South for every 7 miles East. Your average rate of change is -5/7. Vocabulary Average rate of change is the change in y co-ordinates of a function, divided by the change in x co-ordinates. Guided Practice Questions 1) Find the average rate of change of the function f (x) = x 2 as x varies from 1 to 3. 2) If the temperature at 1pm was 82deg, and at 9pm was 50deg, what was the average rate of change of temperature over the given time interval? 3) Brian drove to town to get some milk, he left at 9am, drove 8mi north, 5mi west, 3mi north, and then 2mi west. He arrived at the store at 9:45am. What was his average rate of change of location? 4) If Kelli spends $5 at 1pm, $7 at 2:30pm, $12 at 4pm, and $2 at 4:30pm, what is her average rate of spending? 5) What is the average rate of change shown in the image below? 23

28 1.4. Average Rate of Change Answers 1) The average rate of change is the slope of the line that passes through the two points (3,9)(1,1) on the graph. f (x 2 ) f (x 1 ) x 2 x 1 f (3) f (1) ) The temperature dropped by 32deg over a period of 8 hours: 32 8 = 4 4deg per hour 3) Brian drove a total of 18miles, in a total of 45mins: 18mi 45min = 2 5 miles per minute 4) Kelli spent a total of $26 over a total of 3 1/2 hours: $26 7/2hr $52 7 Multiply top and bottom by 2 to remove the fraction in denominator. $7.43 hr 5) The two points in the image are: ( 6, 1) : (7,6) The average rate of change is the same as the slope of the line. Recall that m = y (6) ( 1) (7) ( 6) 7 13 x y 2 y 1 x 2 x 1 Practice For y = x 3, find the average rate of change as: 1. x increases from 1 to 3 24

29 Chapter 1. Analyzing Functions 2. x increases from -4 to Suppose f (1) = 2 and the average rate of change of f between 1 and 5 is 3. Find f (5) 4. Jamie went on a bicycle trip and stopped regularly at half-hour intervals. At each break he recorded his total distance since leaving home. What was his average speed in km/h, during the first half of the trip? During the last half? Jamie hoped to average at least 11.5km/h over the course of the trip. Did he? Explain. TABLE 1.12: Stops Time (h) Distance (km) On Monday, the price of a gallon of gas was $3.74, and on Saturday, the price had risen to $4.09. What is the average rate of change of the price of a gallon of gas? 6. According to census figures, the population of Clovis was 31,194 in 1980 and 32,511 in What was the average rate of change of the population over that time interval? 7. Griego and Sons will deliver 12 yd 3 of gravel for $240 and 30 yd 3 for $575. What is the average rate of change of the cost as the number of cubic yards varies from 12 to 30? 8. When a load of 5 pounds is placed on a spring, its length is 6 inches, and when a load of 9 pounds is placed on the spring, its length is 8 inches. What is the average rate of change of the length of the spring as the load varies from 5 pounds to 9 pounds? 9. Driving at a speed of 75 mph, a Mini Cooper s fuel efficiency is 29 miles per gallon. If the driver slows to a speed of 60 mph, he will have a fuel efficiency of 34 miles per gallon. What is the average rate of change of the fuel efficiency as the speed drops from 75 mph to 60 mph? 10. Let y = f (x) = x 2 + x + 2 a) Find the average rate of change of y with respect to x between x = 1 and x = 2 b) Draw the graph of f and the graph of the secant line through ( 1, 2) and (2, 4) 11. Amy takes a trip from Chicago to Milwaukee. Due to road construction, she drives the first 10 miles at a constant speed of 20 mph. For the next 30 miles she maintains a constant speed of 60 mph and then stops at McDonald s for 10 minutes for a snack. She drives the next 45 miles at a constant speed of 45 mph. a) What was Amy s average driving speed for the trip? b) What is her average speed for the entire trip (including the stop at McDonald s)? 12. The weight w(t) (in grams) of a tumor t weeks after it forms is given by w(t) = t2 15. Find the average rate at which the tumor is growing during the fifth week after it was formed. 25

30 1.5. Minimums and Maximums Minimums and Maximums Here you will learn to analyze a situation and determine if it is maximization or minimization, write an expression to represent such a situation, and use a graphing utility to find a maximum or minimum. You run a business drawing caricatures. You currently have 25 drawings of rap artists and 25 drawings of "Glee" actors and actresses to sell. You can continue to draw 5 pictures per day if you are not taking the time to sell what you have already. Unfortunately, one of your classmates has seen what a great business this is, and plans to start selling pictures herself. Right now you sell each caricature for $20, but you know when your friend starts to compete for sales, you will have to discount your price to stay competitive. Assuming your sales price goes down by $1 per day, how long should you continue to draw caricatures before selling, so you make the maximum profit? Watch This Embedded Video: Learn how to find minimum and maximum values by watching the video at this link. Guidance Minimums and Maximums In real life, it is common to need to identify what combinations of values result in a maximum or minimum quantity, collectively called extrema. It is important to note that not all functions have extrema. Minimum: Maximum: Formally: The point (c, f (c)) is the minimum value of a function if f (c) f (a) for all elements a (a c) of the domain of f. Informally: The point (c, f (c)) is the minimum if all other function values are greater than or equal to f (c). Formally: The point (c, f (c)) is the maximum of f (x) if f (c) f (a) for all elements a (a c) of the domain of f. Informally: The point (c, f (c)) is the maximum if all other function values are less than or equal to f (c). Example A Determine if each function has a minimum or a maximum point TABLE 1.13: a. y = 2x - 1 b. y = x 4 Solution a. The graph of y = 2x - 1 is a line. It does not have a maximum or a minimum. b. The graph of y = x 4 has a minimum value at (0,0). It does not have a maximum. 26

31 Chapter 1. Analyzing Functions Example B You have 100 feet of fence with which to enclose a plot of land on the side of a barn. You want the enclosed land to be a rectangle. What size rectangle should you make with the fence in order to maximize the area of the rectangular enclosure? Solution: The plot of land will look like the picture below: The area of the rectangular plot is the product of its length and width. We can write the area as a function of x: A(x) = xh. We can eliminate h from the equation if we consider that we have 100 feet of fence, and we write an equation about how we are using that 100 feet of fence: x + 2h = 100. (The fourth side of the rectangle does not require fence because of the barn.) We can solve this equation for h and substitute into the area equation: x + 2h = 100 2h = 100 x h = 50 x 2 A(x) = xh ( = x 50 x ) 2 = 50x x2 2 The graph of A(x) is shown here on the interval [0,100]. Using a maximum function on a graphing utility tells us that the point (50,1250) is the maximum point. This tells us that when the rectangle s width is 50 ft, the area is 1250 ft 2. 27

32 1.5. Minimums and Maximums Example C What is the minimum possible surface area of a box with a square base and a fixed volume of 12 cm 3? Solution: Let the length and the width of the box be x cm, and the height be h cm. We can write the volume equation as x x h = x 2 h = 12. We can also express the surface area in terms of x and h: Surface area = S = 4xh + 2x 2 (The base and the top are squares with area = x 2 and the four sides are each rectangles of area equal to xh ). We can express the surface area as a function of x if we consider the volume equation and the surface area equation as a system of equations: { x 2 h = 12 4xh + 2x 2 = S 28

33 Chapter 1. Analyzing Functions We want to work with the surface area equation since that is what we want to minimize. It will be easier to graph and analyze surface area if we can express S in terms of just one other variable. Rewrite the surface area equation as a function of x: First, rewrite the volume equation: x 2 h = 12 h = 12 x 2 Now, use substitution: S(x) = 4xh + 2x 2 ( ) 12 = 4x x 2 + 2x 2 = 48 x + 2x2 The values of the function S(x) represent different possibilities for the surface area of the box, given that the base is a square, and given that the volume of the box is 12 cm 3. To identify the minimum surface area, we need to find the lowest function values for S(x). The graph below shows the function S(x) on the interval [0,5]. By examining the graph, we can see that the lowest point is between x = 2 and x = 3. If you use a minimum function on a graphing utility, you will find that the minimum point is approximately (2.3, 31.4). This tells us that when the side length of the box is approximately 2.3 cm, the surface area is approximately 31.4 cm 2, which is the smallest it can be. TABLE 1.14: Concept question wrap-up First describe the number of pictures available for sale based on the number of days, starting with the original 50 and increasing by 5 per day: (50 + 5d). Next set the sales price as a function of the number of days, starting at the original price of $20, decreasing by $1 per day: (20 1d). Now multiplying the two expressions together represents the income from the number of pictures available at the current price, based on the number of days from start: (50 + 5d)(20 d) Set the combined function equal to zero and solve for the intercepts: (50 + 5d)(20 d) = 0. This yields the zeroes of -10 and 20. Since the expression describes a parabola, midway between the x coordinates of -10 and 20 would be the vertex, representing the greatest value resulting from the combination of sales price and number of pictures: +5 The greatest profit results from selling the pictures 5 days after the start. 29

34 1.5. Minimums and Maximums TABLE 1.14: (continued) Concept question wrap-up First describe the number of pictures available for sale based on the number of days, starting with the original 50 and increasing by 5 per day: (50 + 5d). Next set the sales price as a function of the number of days, starting at the original price of $20, decreasing by $1 per day: (20 1d). Now multiplying the two expressions together represents the income from the number of pictures available at the current price, based on the number of days from start: (50 + 5d)(20 d) Set the combined function equal to zero and solve for the intercepts: (50 + 5d)(20 d) = 0. This yields the zeroes of -10 and 20. Since the expression describes a parabola, midway between the x coordinates of -10 and 20 would be the vertex, representing the greatest value resulting from the combination of sales price and number of pictures: +5 The greatest profit results from selling the pictures 5 days after the start. If you are curious what the profit would be, or how many pictures would be sold, simply replace the variable (d) with the calculated value of 5. The value of the first expression: [(50 5(5)] represents the number of pictures 5 days along. The value of the second expression [20 1(5)] represents the sales price per picture on day 5. The value of the complete expression: [50 5(5)][20 1(5)] represents total income. Vocabulary Global Minimum: The smallest value of the entire function, symbolically the lowest point on an entire graph. Global Maximum: The greatest value of the entire function, symbolically the highest point on an entire graph. Extrema: The collective term encompassing both minimum and maximum, referring to the "extreme" value of the function in a given direction. Guided Practice Questions 1) In each situation determine if a quantity should be maximized or minimized. a. You have 100 feet of fence to enclose a field, and you want to create the largest field possible. b. You run a factory that packages toilet paper, and you want to use the least amount of plastic possible for each roll. 2) A rectangle has a perimeter of 25in. Write an expression for the area of the rectangle as a function of its width x. 3) Graph your expression from problem #2 4) What dimensions of the rectangle in problem #2 will maximize its area? What is the area? (These values will be approximations) Solutions 1) a) This situation involves maximizing the area of the field. b) This situation involves minimizing the amount of plastic used per roll. (This would be the surface area of a cylinder.) 30 2) The area of a rectangle is l w

35 Chapter 1. Analyzing Functions The perimeter is 2 l + 2 w Therefore we have: 25 = 2(l + w) 12.5 = l + w 12.5 w = l A = x(12.5 x) 3) Using a graphing tool like: - we have: 4) By looking at the graph, we see that when x 6in, the area is 39in 2. Practice 1. What quantity should be maximized? What quantity should be minimized? You are manufacturing chairs, and it costs you a certain amount of money to make each chair. You need to determine the selling price of the chairs. 2. A rectangle has area 20 in 2. Write an expression for the perimeter of the rectangle as a function of its width x. 3. What dimensions of the rectangle in problem #2 will minimize its perimeter? What is the minimum perimeter? (These values will be approximations.) 4. In your own words, define the term maximum of a function. 5. Explain how you can use a graph to identify global extrema of a function. 6. A rectangle has a perimeter of 24 inches. What is the maximum area the rectangle can have? 31

36 1.5. Minimums and Maximums 7. A cylindrical canister has a volume of 30 in 3. What is the radius of the canister with minimum surface area? (Volume of a cylinder is V = πr 2 h 8. Consider the function f (x) = bx For what values of b will the function have a maximum? 9. Consider the function S(x) = 48 x + 2x2. How can you tell that this function does not have a global maximum or minimum? 10. A rectangle has perimeter P. Write a function for the area of the rectangle as a function of P and x, the width of the rectangle. What do you think will be the rectangle with maximum area? 11. A rectangular lot beside a river is fenced on the other 3 sides with 80ft of fencing. What is the largest possible size of the lot? Problems 12-15: Determine whether each function has a maximum or minimum 12. y = x y = x y = x 15. y = x

37 Chapter 1. Analyzing Functions 1.6 Discrete and Continuous Functions Here you will learn to identify discrete and non-discrete functions. You will also explore applications of each. Is a bank account balance a continuous function? How about number of pets per household? Gallons of gas in your car? Number of days that students rode bikes to school in a given week? These are all functions, but they are different types of functions. This Concept is all about learning the difference. Watch This MEDIA Click image to the left for more content. - KeeperOfPhi: Discrete vs. Continuous Relations Guidance Imagine taking a poll to learn the most popular band in school. You interview a large and representative cross-section of students, asking each one how many CD s they have purchased supporting his/her favorite band. You decide that the band with the most CD s sold should be the winner. Wouldn t you be surprised to see numbers like 3.2 or 5.7 on your graph? Who buys.2 or.7 CD s? Now imagine comparing the CD count to the age of the students to see if some bands are more popular with particular age groups. You add birthdate information to your graph. Would you be surprised to see an average student age of 15.4 or 16.7 years? Of course not, you would probably be a lot more surprised to see an average of exactly 15 or 16 years old. The difference between these functions is the topic of this lesson. The number of CD s in the first group is a discrete function, since it is very unlikely that someone would purchase a fraction of a CD. The ages in the second count are a continuous function, since people age constantly and particularly at younger ages keep track more accurately than just "years old". Discrete Functions A discrete function is a function in which the domain and range are each a discrete set of values, rather than an interval in R. Recall from a prior lesson that an interval includes all values between the specified minimum and maximum. If a function is discrete, it does not include all of the values between two given numbers, but rather only specific values in a particular range. Non-Discrete Functions A non-discrete function is one that is continuous either on its entire domain, or on intervals within its domain. The term continuous refers to a function whose graph has no holes or breaks. (Note that this is not a formal definition. To formally define continuity requires that we use the concept of limit, which we will examine in the next lesson. For now it is sufficient to focus on what the graph looks like.) 33

38 1.6. Discrete and Continuous Functions Example A Identify the examples given in the introduction as either discrete or continuous: a) A bank account b) The number of pets in a household c) Age of students taking this class d) Number of days that students rode bikes to school Solution a) The balance in a bank account is counted in dollars and cents, any change is countable and quantifiable. This is an example of a discrete function. b) Discrete function, since one does not generally have a fraction of a pet. c) Continuous function, there is no limit to the level of accuracy you could apply to the age of each student since time is continuous. d) Discrete, the question suggests a specific count of number of days. Example B Identify the function as either continuous or discrete based on the graph: Solution: This graph shows a continuous function, as there are no holes identified on the line, and also no endpoints. Example C Identify the function as either continuous or discrete based on the equation: 34

39 Chapter 1. Analyzing Functions y = x 3 3x Solution The function is continuous, as there is no restriction on the values which may be input for x. Note that this is not a straight line, or even a simple curve like y = x 2. A graph need not be straight or simple to be continuous. Vocabulary A Discrete Function has individual and separated values, the members of the domain may be counted individually. A Continuous Function contains an infinite, uncountable number of values despite the range of the domain, as there is no separation between values. Guided Practice Questions Mark is working at the local fast food restaurant and earns $7.15 per hour. The following table shows the amount of money he earns by working a particular number of hours per week. TABLE 1.15: Hours Worked Money Earned 1 $ $ $ $ $ ) Does this table represent data that is "continuous" or "discrete"? Explain your answer. 2) Write an equation that models the data. 3) Graph your equation. 4) Use the equation or your graph of it to predict what Mark s salary will be if he works 40 hours. 5) If Mark s employer has a policy of rounding up work hours to the nearest 1/2 hour, which of the previous four 35

40 1.6. Discrete and Continuous Functions Answers answers would be affected, and how? 1) The data can be considered continuous because Mark might work any length of time, resulting in any amount of income. 2) Mark earns $7.15 in each hour he works. His income can be represented by: income = hours X $7.15 or y = $7.15x 3) 4) Mark s income after 40 hours will be $ = $ ) If Mark s employer only counts by 1/2 hours, then: a) The equation is no longer continuous, as Mark cannot earn the amounts of money that would be associated with smaller units of time than.5 hours. b) The equation would be unchanged, but would have a domain limited to.5hr increments. c) The graph would look similar, but would be a series of dots rather than a line. d) His salary at 40 hours would remain unchanged, as it was on a.5hr increment already. Practice Identify each of the following variables as being either discrete or continuous The number of telephone calls received at school in a given week. 2. The weight of a bag of oranges. 3. The length of a piece of rope. 4. Speed of a truck. 5. The number of misdemeanor arrests in a town. 6. Number of flaws in a bolt of fabric. 7. The population of the Bald Eagle. 8. A person s age. 9. Does the graph below represent a Continuous or a Discrete Domain?

41 Chapter 1. Analyzing Functions 10. The equation f = 0.305m can be used to convert meters into feet. Is the domain of this function discrete or continuous? 11. Your local gardener tells you that your corn plant will grow 1.25 taller each month. It is now 6 tall. Write a formula that will tell you how tall your plant is at any time in the future. Is there a continuous or a discrete domain? 12. You can buy T-shirts for $12.00, or hats for $ Write an equation showing how much you will spend (y) for any combination of hats (h) and t-shirts (t) that you purchase. Is the domain discrete or continuous? For questions 13-15: A local neighborhood homeowners association is asking the community residents to participate in a recycling initiative. At the end of each week, each resident is asked submit the number of plastic containers they recycle to the HOA. The data collected was compiled into the following table: TABLE 1.16: House # Plastic Containers Does this table represent data that is "continuous" or "discrete"? Explain your answer 14. Why can you not really write an equation to model the data? 15. Can you predict how many plastic containers the 6th house on the block will recycle the next week? 37

42 1.7. Increasing and Decreasing Functions Increasing and Decreasing Functions Here you will learn about functions which increase or decrease in value from left to right. You will also explore functions which change along the way, increasing some times and decreasing others. How do you describe the behavior of a function? One useful way is to identify it as increasing or decreasing, meaning the graph goes up or down from left to right. What about graphs that are not straight lines? What if they increase and decrease in different places on the same graph? Watch This Embedded Video: MEDIA Click image to the left for more content. - James Sousa: Determine Where a Function is Increasing and Decreasing Guidance In this lesson we will consider the function values in between extrema (extremes: minimums or maximums), focusing on where function values are increasing or decreasing. Formally: A function is increasing on some interval of its domain if f(a) >f(b) for all a, b in that interval such that a >b. A function is decreasing on some interval of its domain if f (a) <f (b) for all a, b in that interval such that a >b. Informally: A function is increasing on a section if the graph of that section rises to the right. A function is decreasing on a section if the graph of that section falls to the right. Example A Describe the behavior of the graph below, which shows the federal minimum wage from 1975 to 2005: 38

43 Chapter 1. Analyzing Functions Solution: Overall, from 1975 to 2005, the minimum wage (in dollars per hour) increased from $2.10 per hour to $5.15 per hour. However, if we examine the graph on smaller intervals, we can see that this increase was not steady. For example, the minimum wage increased from $2.30 in 1977 (year 3) to $3.35 in 1981 (year 7), and then it was constant for several years at $3.35 per hour. We can describe this situation using intervals. For example, we can say that the function increases on the interval (3,7) and is constant on the interval (7, 15). Example B The graph below shows the minimum wage data adjusted for inflation. (Since prices for food and other necessities tend to go up over time, comparing absolute dollar amounts doesn t make sense. The adjustment shows how much one can buy with minimum wage.) State the intervals on which the graph increases and the intervals on which it decreases. Use proper interval notation. Solution: The function decreases on the intervals (2, 3), (5, 15), (18, 21), (24, 31). The function increases on the intervals (1, 2), (3, 4), (15, 17), (22, 24). Notice that this is a discrete function, and the intervals are ranges of specific points. That is, the function is decreasing if, when we look at the graph, the points are sloping down from left to right. The function is increasing if the points are sloping up from left to right. This will be the case for any discrete function. 39

44 1.7. Increasing and Decreasing Functions Example C Identify the intervals of increase and decrease, use interval notation: y = x 3 3x Solution: The function increases on the interval (, 1) and on the interval (1, ). The function decreases on the interval ( 1,1). These are open intervals (with parentheses instead of brackets) is because the function is neither increasing nor decreasing at the moment it changes direction. We can imagine a ball thrown into the air. The height increases up to a maximum point before it starts to decrease. What happens at that maximum point? There is an instant when the height is not changing; it is neither increasing nor decreasing, so those specific values are not included on the interval. TABLE 1.17: Concept question wrap-up: Graphs with curves or which go up and down throughout the graph may be described with the use of intervals. By splitting the description of the graph into sections, each with consistant behavior throughout the section, the entirety of the graph may be evaluated. Vocabulary An increasing function is one with a graph which goes up from left to right. A decreasing function is one with a graph which goes down from left to right. An interval is a specific and limited part of a function. Guided Practice Questions 40 1) State the intervals on which each function increases and decreases. a) y = x 2 b) y = x

45 Chapter 1. Analyzing Functions 2) Give an example of a discrete function that increases and decreases on different intervals of its domain. 3) State the intervals on which the function increases and decreases. 4) State the intervals on which the function increases and decreases. 5) Kim earns tips at work on Mon, Wed, and Fri. She likes to go to the movies with her friends on Tues evenings when there are no crowds. On Saturdays, she and her boyfriend go skating in the park and always stop for ice cream. Describe the intervals where Kim s money is increasing and decreasing. 41

46 1.7. Increasing and Decreasing Functions Answers 1) a) y = x 2 The graph of this function is shown here. The function decreases on (,0) and it increases on (0, ). b) y = x The graph of this function is shown below. This function is a line with positive slope. That is, it rises to the right. Therefore the function is increasing on its entire domain, R. 2) Answers will vary, but should describe a function such as a person s height compared to his or her age. That is, values will increase until the age at which the person stops growing. Then the function will be constant for several decades. Then the function will decrease slightly, as some people lose height in their later years. 3) The function in the image is linear, and has no breaks or holes, it is also rising to the right. This function is increasing throughout its domain. 4) The function in the image is falling to the right, or decreasing between x = -5 and approximately x = 2. At x = 2, the function begins to increase (rise to the right), and continues to increase until the endpoint. 5) Kim s money is increasing for the interval between monday and tuesday evening, decreasing for the tuesday to wednesday interval, increasing during the interval between work on wednesday and ice cream on saturday, and finally decreasing between ice cream on saturday and work on monday. Practice Questions For questions 1 and 2, consider the following graph: 42

47 Chapter 1. Analyzing Functions 1. Approximate the coordinates of the relative maximum of the graph 2. Approximate the coordinates of the relative minimum of the graph. 3. Consider the equation of a linear function y = mx, where m is the slope of the line. For what values of m will the function be an increasing function? For what values of m will the function be a decreasing function? 4. Consider the graph above for questions 2 and 3. On what intervals is the function increasing? On what intervals is the function decreasing? 5. Sketch a possible graph of the function described here: The function is continuous on R. It is decreasing on the interval (,2) and increasing on the interval (2, ). 6. Explain in your own words the difference between relative and global extrema. 7. For the function shown in the graph below, give approximate coordinates of all global and relative extrema. 8. Consider the function f (x) = x x. Use a graphing utility to sketch a graph of this function and to calculate a relative minimum. 9. A rectangle is inscribed in a semi-circle of radius 3. a) Write an equation to represent the area of the rectangle as a function of x. b) Graph the equation and calculate the maximum area of the rectangle. 10. How many relative extrema should there be on a graph of y = x 6? 11. Sketch a graph of a function which has an absolute maximum, but no local maximum, on the interval [-5, 7]. 12. Sketch a graph of a function with two local minimums on the interval (-4, 9], and no global extrema. Questions 13-15: Sam makes $7.50 per hour at work, and he works anywhere from 15-40hrs per week. Occasionally the store gets really busy, and Sam s boss allows him to work up to 15hrs of overtime. Sam loves the busy weeks, because he makes "time-and-a-half", or $11.25, for overtime hours. 13. Sketch a graph showing Sam s regular income range. Where are the local extrema? 14. What happens to the local extrema if Sam is working a week with overtime? Where are they now? 15. Sketch another graph including Sam s regular and overtime income, identifying all local extrema. 43

48 1.8. Limits and Asymptotes Limits and Asymptotes Here you will learn about the end behavior of functions. End behavior is commonly investigated when working with very large and very small values of x. In this lesson, you will also learn about asymptotes, lines on a graph to which a progressive value comes very, very close (infinitely close, in fact), but never actually contacts. Suppose you stand exactly 4 feet from a wall, and begin moving toward the wall by halving the distance remaining with each step. How many steps would it take to actually get to the wall? How far would you walk in the process? This is a question of limit, see if you can puzzle it out before the review at the end of this lesson. Watch This Embedded Video: MEDIA Click image to the left for more content. - Khan Academy: Introduction to Limits (HD) Guidance Limits and Asymptotes Consider the function f (x) = 1 x. A graph of this function is shown below. 44

49 Chapter 1. Analyzing Functions Notice that as the values of x get larger and larger, the graph gets closer and closer to the x-axis. In terms of the function values, we can say that as x gets larger and larger, f (x) gets closer and closer to 0. Formally, this kind of behavior of a function is called a limit. We say that as x approaches infinity, the limit of the function is 0. The line y = 0 is called the asymptote of the graph, it represents the value that f (x) will never quite reach. We can also say that f (x) = 1 x is asymptotic to the line y = 0. If we consider the behavior of the function as x approaches, we see the same result: the limit of the function has x approaches is also 0. Notice that this has the same asymptote: y = 0. To be even more formal, we can write limits using a special notation. For the first limit, we write: lim x f (x) = 0. For the second limit, we write lim x f (x) = 0. For any limit, here we will always write the x under the abbreviation lim, and then we will write the function under consideration. We can also write each of these limits with the specific function: lim x 1 x = 0 and lim x 1 x = 0. Because we are focused on end behavior, we are considering the limit of functions as x approaches ±, and so the asymptotes we will find are horizontal lines. If we were examining other aspects of functions, we might find asymptotes that are vertical lines. For example, the function f (x) = 1 x has a vertical asymptote at x = 0, or the y-axis. That is, the graph approaches the y-axis, as x values get closer and closer to 0. Example A Write the limit described using limit notation. The limit of some function f (x) as x approaches infinity is 2. Solution: We write the limit as follows: lim x f (x) = 2 Example B Explain in words the meaning of the limit statement: lim x ( x ) = 3 Solution: lim x ( x ) = 3 means: "As larger and larger numbers are substituted in for x in the function 3 + 2/x, the value comes closer and closer to 3. This is due to the fact that the value added to 3 gets smaller and smaller, down to effectively 0 as 2 is divided by larger and larger numbers. Example C Determine the horizontal asymptote of the function g(x) = 2x 1 x and express the asymptotic relationship using limit notation. Solution: This function is asymptotic to the line y = 2. The limit is written as lim x 2x 1 x = 2. We can determine the asymptote (and hence the limit) if we look at the graph. However, we can also analyze the equation to determine the limit. Consider the function g(x) = 2x 1 x. As x approaches infinity, the x values are getting larger and larger. For sufficiently large values of x, the values of the expression 2x - 1 are very close to the values of the expression 2x, because subtracting one from a large number is fairly insignificant. Thus for sufficiently large 45

50 1.8. Limits and Asymptotes values of x, 2x 1 x 2x x 2. As you can see from the accompanying table, which was created by a TI-83 graphing calculator, the function value gets closer to 2 as we look at larger and larger x values. TABLE 1.18: You will recall the question at the beginning of the lesson about the distance involved in a strange walk towards a wall. "If you start 4ft from a wall, and halve the distance to it with each step, how many steps will it take, and how far will you walk, before you actually touch the wall?" Logically, we know that there is only a total distance of 4 feet between you and the wall, so no matter how you break it up, you cannot walk more than 4 feet. However, the actual distance you cover, and the number of steps it would take, cannot truly be defined since there could always be 1/2 of the remaining distance left. Technically speaking, you could continue the process forever without actually touching the wall! Of course, in practice, your ability to only move 1/2 of the remaining distance is limited by muscle control and measurement accuracy, so you would touch the wall before ( very ) many steps were actually taken. Mathematically: lim n = 4 Where n is the number of steps. n In other words: "As the remaining distance gets closer and closer to 0, the total distance approaches 4" Vocabulary A limit is a value which the output of a function never reaches, but progresses toward as larger or smaller inputs are evaluated. An asymptote is a line representing a limit on a graph, visually demonstrating the value which cannot actually be arrived at by the output. End behavior is a description of the trend of a function as input values become very large or very small, represented as the ends of a graphed function. Guided Practice Questions Describe each case given below and sketch a graph of each function with the given properties: 1) lim x f (x) = 2 2) lim x f (x) = 0 3) lim x 0+ f (x) = 4) lim x 4 f (x) = 5) lim x 4 f (x) = 3 Answers 46

51 Chapter 1. Analyzing Functions There are a number of possible graphs for each of the required cases above, one example of each is offered below. 1) This reads: "The limit of f(x) as x approaches infinity is 2." In other words, as x gets very big, f(x) or y gets infinitely close to 2. 2) This reads: "The limit of f(x) as x approaches negative infinity is 0." In other words, as x gets massively negative, f(x) or y gets infinitely close to 0. 3) This reads: "The limit of f(x) as x approaches zero from the positive direction is infinite." In other words, as x approaches from the right side of the graph, and gets infinitely close to zero, f(x) or y grows infinitely large. 47

52 1.8. Limits and Asymptotes 4) This reads: "The limit of f(x) as x approaches 4 is negative infinity." In other words, as x gets infinitely close to 4, f(x) or y gets infinitely negative. 48 5) This reads: "The limit of f(x) as x approaches 4 is 3." In other words, as x gets infinitely close to 4, f(x) or y gets infinitely close to 3. This can be a straight line, as y approaches 3 when 4 approaches 4 from either direction.

53 Chapter 1. Analyzing Functions Practice 1. Define the terms horizontal asymptote and vertical asymptote. 2. Explain the difference betweenlim x 6 f (x) = and lim x f (x) = 6 3. Explain whatlim x f (x) = 200 means 4. Explain whatlim x 175 f (x) = 175 means Evaluate the following limits, if they exist. If a limit does not exist, explain why. 3t 5. lim 2 7t t t 8 6. lim t 3 7. lim t (t 2 t 4 ) 8. lim x x + x 2 + 2x 9. Find the horizontal and vertical asymptotes of the following function: h(g) = 5g2 7g+9 g 2 2g 3 Given: f (x) = x2 x 6 perform the following: x 2 2x Find the horizontal and vertical asymptotes. Determine the behavior of f near the vertical asymptotes. 11. Find the roots, y intercept and holes in the graph. Determine lim t 1 t n if: 12. n > n < n = 0 Let G & H be polynomials. Findlim x G(x) H(x) if: 49

54 1.8. Limits and Asymptotes The degree of G is less than the degree of H 16. The degree of G is greater than the degree of H 17. The degree of G is the same as the degree of H 18. A pool contains 8000 L of water. An additive that contains 30g of salt per liter of water is added to the pool at a rate of 25 L per minute. a) Show that the concentration of salt after t minutes in grams per liter is: C(t) = (t)30g l+25(t)l b) What happens to the concentration as time increases to? Physically, why does this make sense? 50

55 Chapter 1. Analyzing Functions 1.9 Infinite and Non-Existent Limits Here you will explore the concept of functions which approach infinity or negative infinity. You will also learn about functions which do not approach a specific value. You have heard time and again that it is "against the rules" to divide by the number 0. Even the most basic calculator will return some form of "ERROR" if you try to divide even the smallest of numbers by 0. Do you really understand why it is "against the rules"? What is really wrong with dividing by nothing? Watch This Embedded Video: MEDIA Click image to the left for more content. - James Sousa: Limits at Infinity. Guidance Infinite limits Functions can exhibit a number of different behaviors as the input value gets very large or very small. As x approaches, some functions output values closer and closer to a single number, some approach zero, and some continue to get larger and larger or smaller and smaller without limit. In this lesson, we will explore functions of the last type, functions with infinite limits, and the different types of asymptotes they may have. Example A Evaluate the function h(x) = x2 2x 1. 51

56 1.9. Infinite and Non-Existent Limits Solution To evaluate this function, consider the behavior of the function as larger and larger values are Inserted for x. As x approaches, the function values also approach. Therefore the limit of the function as x approaches is: lim x x 2 2x 1 =. Similarly, as x approaches, f (x) approaches. Therefore we have lim x x2 2x 1 =. We can also understand this limit if we analyze the equation for h(x). As x gets larger and larger, the value of the expression 2x - 1 gets closer and closer to the value of the expression 2x. That is, for sufficiently large values of x, 2x x. Therefore the values of h(x) approach x2 2x = x 2. As x gets larger and larger, so does x 2. For large values of x, the function h(x) gets closer and closer to x 2. Therefore the limit is infinity. Example B Approximate the function f (x) = x 2 + 2x - 3. Solution This function has an infinite limit as x approaches infinity. However, as x gets larger and larger, f (x) x 2, since the x 2 value grows much more quickly than the 2x value, particularly apparent at very large +/- values of x. If this is not immediately apparent, evaluate the function for x = 1,000,000, and you will quickly get the idea! Therefore we can use the function y = x 2 to describe the end behavior of f (x). Example C Describe the end behavior of each graph. That is, determine if the function has a limit L, if the limit is infinite, or if the limit does not exist. a.) y = x 2 b.) y = 2( 1) x c.) y = 1 1 x Solution: a.) y = x 2 As x approaches +, x 2 also approaches +. As x approaches, x 2 approaches +. Therefore lim x x 2 = lim x x 2 =. b.) y = 2( 1) x This function is difficult to understand without producing a graph. The table shows that the function only takes on two values: 2, and -2, and is undefined at non-integer values of x. As x approaches + or, the function values 52

57 Chapter 1. Analyzing Functions alternate between 2 and -2. Therefore the limit does not exist. c.) y = 1 1 x If you look at the table of this function, which has negative and positive values of x, you can see that as x approaches + or, the function values approach 1. ) ( ) Therefore lim x (1 1 x = lim x 1 1 x = 1. We can also determine this limit analytically. For large values of x, x is also large, and so 1 x is small (since dividing 1 by a large number results in a very small number). Therefore, for large values of x, 1 1 x 1 0 = 1. We can make the same argument for x approaching. TABLE 1.19: Concept question wrap-up Dividing by zero is "against the rules" because there is no definition for the answer you would get. Consider what happens as you take a given value and divide it by smaller and smaller numbers: 2 / 10 = 1/5 2 / 1 = 2 2 /.1 = 20 2 /.001 = 2,000 2 / = 2,000,000,000 As we divide by progressively smaller numbers, the quotient gets larger and larger. Also, we can see that in each case, the problem could be reversed by multiplying the product by the dividend to get the divisor, for instance: 2 /.1 = 20 ==>20 *.1 = 2. Unfortunately, this doesn t work if you actually divide by 0, even if you assume that dividing by zero resulted in infinity! No matter how big the number you multiply by zero, even infinity, you will never be able to get back to 2. x/0 = unde f ined 53

58 1.9. Infinite and Non-Existent Limits Vocabulary An infinite function is one whose output approaches infinity or negative infinity as very large or very small values are calculated for the input variable (usually "x"). An asymptote is a line representing a value toward which the value of a function may approach, but never actually reach. Guided Practice Questions x 1) Evaluate lim 3 x x 7 x 2) Evaluate lim 2 +2x x x 3 3) Evaluate lim x x 3 4) Describe the end behavior of y = 2x 2 + 3x 7 5) Evaluate 2x 3 5x 2 + 8x Solutions 1) The degree of the numerator is greater than the degree of the denominator, so the function will grow without bound. Since the denominator is x - 7, the function cannot include x = 7, because the function cannot be defined where the denominator = 0. Logically, as x gets huge, the -7 matters less and less, and we end up with just x 2. 2) Similar to the last problem, the numerator is of greater degree than the denominator, so the function does not approach a limit. The denominator is x - 3, so the graph cannot include 3. 3) As x, or in other words, "as x gets huge", the value of x 3 grows even faster, either positive or negative, so there is no limit. 4) As x grows huge, x 2 grows much faster than the rest of the expression, therefore, we can approximate the end behavior of 2x 2 + 3x 7 with y = x 2 5) 2x 3 5x 2 + 8x is a 3rd degree equation, so it will turn twice, since it is not a rational function, there are no concerns about numerator or denominator. The function will have no limits, and will grow without bound in both the positive and negative directions. If you use a graphing calculator to graph the function, you will see that y = x 3 can be used to approximate it. Practice Problems 1-3: The limit of f (x) as x t cannot exist if which conditions are true? List three conditions. 4. Give an example of a limit that does not exist Problems 5-7: Assuming that f (x) is a rational function: 5. What is lim x f (x) when the degree of the numerator is less than the degree of the denominator? 6. What is lim x f (x) when the degrees of the numerator and the denominator are equal? 7. What is lim x f (x) when the degree of the numerator is greater than the degree of the denominator? In general, if r is a positive real number, what is lim x 1 x r? 9. In general, if r is a positive real number, what is lim x x r?

59 Chapter 1. Analyzing Functions Problems 10-13: Let a and b be real numbers and let t be a positive integer. Complete each of the following properties of limits. 10. lim x a x t = 11. If f is a polynomial, lim x a f (x) = 12. lim x a k f (x) = 13. lim x t x = Problems 14-16: Evaluate. 14. lim x 5 (5+x) 2 25 x 15. lim x 3 x 3 6x+2 x 2 +2x lim x 0 (3+3y) 1 3y 1 x 55

60 1.10. Linear and Absolute Value Function Families Linear and Absolute Value Function Families Here you will learn about function families, and about two kinds of function families in particular: Linear Functions and Absolute Value Functions. On Tues., Mr. Varner s math class filed into the room, and gawked at the message on the whiteboard: "The first student to add together all of the numbers between 1 and 100 wins four free movie tickets to the theater next Friday!" Everyone grabbed a pencil and started adding: No one was further than about 20 when Brian walked in, late as usual, looked at the white board for about 15 secs, and wrote: "5050" on the bottom. The surprised Mr. Varner handed Brian the tickets and told him to take his seat. How did he come up with the answer so fast? Watch This MEDIA Click image to the left for more content. - Coach McCullers: Function Families Guidance In this Concept we will examine several families of functions. A family of functions is a set of functions whose equations have a similar form. The parent of the family is the equation in the family with the simplest form. For example, y = x 2 is a parent to other functions, such as y = 2x 2-5x + 3. Linear Function Family An equation is a member of the linear function family if it contains no powers of x greater than 1. For example, y = 2x and y = 2 are linear equations, while y = x 2 and y = 1 x are non-linear. Linear equations are called linear because their graphs form straight lines. As you may recall from your earlier studies of algebra, we can describe any line by its average rate of change, or slope, and its y-intercept. (In fact, it is the constant slope of a line that makes it a line!) These aspects of a line are easiest to identify if the equation of the line is written in slope-intercept form, or y = mx + b. The slope of the line is equal to the coefficient m, and the y-intercept of the line is the point (0, b). Note that a line can be a member of a family such as the family of "linear functions", and also a member of a sub-family of linear functions with the same slope. The graphs of this subfamily will be a set of parallel lines. One particular subfamily of linear functions is the constant function subfamily. The line x = 5 is a constant function, as the function values are constant, or unchanging. The constant functions sub-family of linear functions is composed of functions whose graphs are horizontal lines. Absolute Value Function Family 56

61 Chapter 1. Analyzing Functions Let s first consider the parent of the family: y = x. Because the absolute value of a number is that number s distance from zero, all of the function values of an absolute value function will be non-negative. If x = 0, then y = 0 = 0. If x is positive, then the function value is equal to x. For example, the graph contains the points (1, 1), (2, 2), (3, 3), etc. However, when x is negative, the function value will be the opposite of the number. For example, the graph contains the points (-1, 1), (-2, 2), (-3, 3), etc. As you can see in the graph below, the absolute value function forms a V shape. There are two important things to note about the graph of this kind of function. First, the absolute value graph has a vertex (a highest or lowest point) and a line of symmetry (a line that splits the function into equal and opposite halves ). For example, the graph of y = x has its vertex at (0, 0) and it is symmetric across the y-axis. Second, note that the graph is not curved, but composed of two straight portions. Every absolute value graph will take this shape, as long as the expression inside the absolute value is linear. Piece-wise Defined Functions Consider again the function y = x. For positive x values, the graph resembles the identity function y = x. For negative x values, the graph resembles the function y = -x. We can express this relationship by defining the absolute value function in two pieces: { x,x < 0 f (x) = x,x 0 We can read this notation as: the function values are equal to -x if x is negative. The function values are equal to x if x is 0 or positive. A piece-wise defined function does not have to represent a function that can already be written as a single equation, such as the absolute value function. For example, one piece may be from one function family, while another piece is from a different function family. Example A Identify the slope and the y-intercept of each line: TABLE 1.20: a. y = 2 3 x 1 b. y = 5 c. x = 2 d. y = 2 3 x + 3 Solution: 57

62 1.10. Linear and Absolute Value Function Families a. y = 2 3 x 1 This line has slope (2/3) and the y-intercept is the point (0, -1). b. y = 5 This is a horizontal line. The slope is 0, and the y-intercept is (0,5). c. x = 2 This is a vertical line. The slope is undefined, and the line does not cross the y-axis. (Note that this line is not a function!) d. y = 2 3 x + 3 The slope of this line is 2/3, and the y-intercept is the point (0, 3). Example B Graph the following: y = 2x 1 and y = 2x 2 1 Solution The graph of y = 2x 1 makes a V shape, much like y = x The function inside the absolute value, 2x+1, is linear, so the graph is composed of straight lines. The graph of y = 2x 2 1 is curved, and it does not have a single vertex, but two cusps. The function inside the absolute value is NOT linear, therefore the graph contains curves. Example C Sketch a graph of the function { x 2,x < 2 f (x) = x + 3,x 2 Solution: 58

63 Chapter 1. Analyzing Functions It is important to note that the pieces of a piece-wise defined function may or may not meet up. For example, in the graph of f (x) above, the function value is 4 at x = -2, but the piece of the graph that is defined by x + 3 is headed to the y value of 1. Therefore the two pieces do not meet. TABLE 1.21: Concept question wrap-up Brian recognized that he didn t need to add each of the numbers individually, only the pairs: = 100, so does , and , and so on. Since there are 50 pairs of 100, that adds up to The only number without a pair is 50, so it gets added to the total: Function families represent this same sort of time-saver. By recognizing common bits of information and combining them in different ways, we can automate some very complex-seeming processes. There is a (very) detailed exploration of this trick at e-numbers-1-to-100/. Vocabulary A linear equation always graphs a straight line, it is characterized by an x term with a power of 1. An Absolute Value Equation generally graphs a V shape, and is in the form: y = x. Piece-wise defined functions are formed from pieces of other families, and my have breaks, or locations on the graph which are not defined. Function families are groups of functions with similarities which make them easier to graph when you are familiar with the parent function, the most basic example of the form. Guided Practice Questions 1) Find the x-intercepts of the function: f (x) = 8 x ) What is the graph of y = x? How is that graph related to the graph of y = a x h +k? 3) What happens to the graph of y = x when the equation changes to y = x 5? 4) What is the y-intercept of the graph of the function: f (x) = 2 x ) If the graph of f (x) = x is shifted three units to the right, then what would be the equation of the new graph? Answers 1) To find the x-intercepts, set f(x) equal to 0, and solve for x: 59

64 1.10. Linear and Absolute Value Function Families 0 = 8 x = 8 x 7 8 = x 7 8 = (x 7) or 8 = (x 7) 15 = x or 1 = x the x-intercepts are 15 and -1 2) The graph of y = x is shown below. You can either use a graphing tool, or plot points, noting that every positive x has a matching y, and every negative x matches with its positive equivalent as y. y = x is the simplest example of a graph in the absolute value function family, of which y = a x h +k is the parent. Changes to a, h, and k shift the graph of y = x in different ways. 3) The graph of y = x is above, the graph of y = x 5 is below. It is clear that the -5 after the absolute value causes the graph to shift down 5 places. The link: Graph ofy = x + k takes you to a graph of y = x + k that has a slider for k, allowing you to see the effect of changing the value of a number added or subtracted after the absolute value. 60 4) To find the y-intercept of f (x) = 2 x solve the equation for x = 0. y =

65 Chapter 1. Analyzing Functions y = y = 12 5) To shift the graph of an absolute value equation horizontally, add or subtract from inside the absolute symbol. f (x) = x + 5 shifts the graph 5 places left, as we see above. f (x) = x 3 would shift the graph 3 places right. Practice For questions 1-5, identify the family that each function belongs to: 1. y = x 7 2. y = 3x 4 3. f (x) = x 2 4. x 2 = y 5. f (x) = x + 3x 2 61

66 1.10. Linear and Absolute Value Function Families { x,x 0 6. Graph the following piecewise function by hand: f (x) = x,x < 0 7. On your graphing calculator, graph the function f (x) = x 2, and answer the following questions: a) What is the shape of the graph? b) Compare the graph to the graph in the problem above. What is the difference between the two graphs? c) What is the slope of the two lines that create the graph? For each equation that follows, identify the coordinates of the vertex of the graph, without actually graphing. 8. f (x) = 6x 9. f (x) = x f (x) = x f (x) = x The graph of p(x) = x is shown below. If t(x) = x 3, how will the graph of t(x) be different from the graph of p(x)? 13. Graph the absolute value equation, create your own table to justify values: f (x) = x Graph the absolute value equation, create your own table to justify values: g(x) = x + 3 Identify the parent function for each set of linear functions. Graph each set of functions using a graphing calculator. Identify similarities and differences of each set. 15. a) f (x) = x 7 b) f (x) = x 2 c) f (x) = x + 1 d) f (x) = x + 5 e) f (x) = x + 10 Parent Function: Similarities: Differences: 16. a) f (x) = 2 11 x b) f (x) = 1 2 x c) f (x) = 2 3x Parent Function: Similarities: Differences: 17. a) f (x) = x 7 b) f (x) = 2x c) f (x) = 4x d) f (x) = 2x + 5 e) f (x) = 6x 10 Parent Function: Similarities: Differences: Use the standard form of a linear equation: f (x) = ax + b and your investigations above to help you answer the following questions: 18. How does the a value affect the graph? 19. How does the b value affect the graph? 20. How are the domain values similar/different? 21. How are the range values similar/different? 22. Does the a and/or b value affect the domain? 23. Does the a and/or b value affect the range? 62

67 Chapter 1. Analyzing Functions 1.11 Square and Cube Function Families Here you will learn about evaluating more complex functions involving powers and roots by identifying the family each function belongs to in order to simplify the general form of the function s graph. Often, the most challenging part of completing a math problem is just getting started on the right track. Once you have a feel for how a particular problem should be solved, crunching the numbers is generally not very difficult. An understanding of function families can be a big help with this, as it gives you an idea of what a more complex function should look like once it has been graphed, just by identifying the parent or most simplified version of the function. Do you know the parent functions for the Square, Cube, Square Root, and Reciprocal function families? Watch This Embedded Video: MEDIA Click image to the left for more content. TMatsu31: IB Math Section 5A Families of Functions Guidance Function Family: Square Functions A square function is a 2nd degree equation, meaning it has an x 2. The graph of every square function is a parabola. A parabola has a vertex, and an axis of symmetry. The graph below shows these aspects of the graph of y = x

68 1.11. Square and Cube Function Families Function Family: Cubic Functions A cube function is a third-degree equation: x 3 and which does not contain negative or fractional exponents. In general, the graphs of cube functions have a particular shape, illustrated by the graph shown here: Cubic functions have a similar shape. However, only some cubic functions will have a relative maximum and minimum. For example, the graph of y = (x - 2) 3-5x shown above, has a relative maximum around x = 0.7, and a relative minimum around x = 3.3. The shape of the cubic graph means that we can predict end behavior: one end will approach, and the other will approach. It is important to note here that the cubic function grows faster than an associated quadratic function. For example, y = x 3 grows faster than y = x 2. Function Family: Square Root Functions Consider the parent of the family, y = x. The domain of the function is limited to real numbers 0, as the square root of a negative number is not a real number. Similarly, the range of the function is limited to real numbers 0. This may seem confusing if you think of squares having two roots. For example, 9 has two roots: 3, and -3. However, for y = x, we have to define the function value as the principal root, which means the positive root. 64 The function y = x is shown below:

69 Chapter 1. Analyzing Functions The same kind of limitations of domain and range will exist for any square root function. Function Family: Reciprocal Function The function y = 1/x has a rather surprising graph. First, the domain cannot include 0, as the fraction (1/0) is undefined. The range also does not include 0, as a fraction can only be zero if the numerator is zero, and the numerator of y = 1/x is always 1. In order to understand what these limitations mean for the graph, we will consider function values near x = 0 and y = 0. First, consider very small values of x. For example, consider x = This yields y = 1/x = (1/0.001) = As we get closer and closer to x = 0, the function values approach. On the other side of the x-axis the function values will approach. We can see this behavior in the graph as a vertical asymptote: the graph is asymptotic to the y-axis. We can also see in the graph that as x approaches + or, the function values approach 0. The exclusion of y = 0 from the range means that the function is asymptotic to the x-axis. Example A Describe the end behavior of each function, and identify the parent function for each: 65

70 1.11. Square and Cube Function Families TABLE 1.22: a. y = x 2-1 b. y = -x Solution: The parent of both functions is y = x 2 a. y = x 2-1 The graph of this function is a parabola that opens up. Therefore lim x ± (x 2 1) =. b. y = -x The graph of this function is a parabola that opens down. Therefore lim x ± (x 2 1) =. All square functions have either a global maximum or minimum. The location of the maximum or minimum is always the vertex of the parabola. Square functions also share behavior in terms of their average rate of change. Consider for example the functions f (x) = x 2, g(x) = x 2-3, and g(x) = 2x 2-3. The table below shows the average rate of change (ARC) of each function on several intervals. Note: The ARC of a function on the interval (a, b) is f (b) f (a) b a. TABLE 1.23: Interval ARC of f (x) ARC of g(x) ARC of h(x) (-1, 0) (0, 1) (1, 2) (2, 3) Notice that the average rate of change of f (x) and g(x) is the same on each interval, and the average rate of change of h(x) is twice that of the other two functions. You may also notice that the average rate of change follows a linear pattern: on each interval the rate increases at a constant rate of 2. While linear functions have a constant rate of change, quadratic functions have an average rate of change that follows a linear pattern. Example B Graph the function y = 3 x + 1, identify the parent function, and state the domain and range of the function. Solution: The parent function is y = x 66

71 Chapter 1. Analyzing Functions From the graph you can see that the function does not take on any x values above 3. (Why not?) Therefore the domain is limited to real numbers 3. The function s lowest value is 1, so the range is limited to all real numbers 1. It is important to note that while the graph of a square root function might look as if it has horizontal asymptote, it does not. The function values will grow without bound (though relatively slowly!). Example C Graph the function f (x) = 2/(x - 3), identify the parent, and identify horizontal and vertical asymptotes. Solution: The parent function is y = 1/x The graph is asymptotic to the x-axis (y = 0) and to the line x = 3. TABLE 1.24: 67

72 1.11. Square and Cube Function Families TABLE 1.24: (continued) Can you identify the parent functions of the square, cube, square root, and reciprocal functions now? Square: y = x 2 Cube: y = x 3 Square Root: y = x Reciprocal: y = 1/x Learning the function families is one of the fastest way to graph complex equations. Using parent functions and transformations (which are detailed in another set of lessons), you can graph very complex equations rather easily. Vocabulary A set of functions which share a similar type of graph is called a Function Family. Transformations are used to change the graph of a parent function into the graph of a more complex function. A principal root is the positive root of a number. The graph of a quadratic function is usually a parabola (a rounded "V" shape). A parabola has a vertex, which is the maximum or minimum value of the function, and an axis of symmetry which separates the two identical halves of the parabola. Guided Practice Questions Identify the parent function within each set of functions. calculator. Identify similarities and differences of each set. Graph each set of functions using a graphing TABLE 1.25: f (x) = x 2 10 f (x) = x 2 1 f (x) = x 2 f (x) = x f (x) = x ) a) Parent Function: b) Similarities: c) Differences: TABLE 1.26: f (x) = 1 8 x2 f (x) = 3 5 x2 f (x) = 9 10 x2 f (x) = x 2 f (x) = 3x 2 f (x) = 8x 2 2) a) Parent Function: b) Similarities: c) Differences: 68 TABLE 1.27: f (x) = (x + 9) 3 f (x) = (x + 2) 3 f (x) = x 3 f (x) = (x 4) 3 f (x) = (x 8) 3

73 Chapter 1. Analyzing Functions 3) a) Parent Function: b) Similarities: c) Differences: TABLE 1.28: f (x) = 1 16 x3 f (x) = 1 2 x3 f (x) = x 3 f (x) = 5x 3 f (x) = 10x 3 4) a) Parent Function: b) Similarities: c) Differences: Solutions 1) This set of graphs can be seen at: Graph for Q#1 a) The parent function of this group of quadratic functions is the most basic function in the set: f (x) = x 2 b) Similarities in this set include: width, shape, end behavior, and degree c) Differences in this set include: x and y intercepts 2) This set of graphs can be seen at: Graph for Q#2 a) The parent function in this group of quadratic functions is the same as the last: f (x) = x 2 b) Similarities include: width, domain, x and y intercept, shape, and degree 3) Differences include: direction, range, end behavior 3) This set of graphs can be seen at: Graph for Q#3 a) The parent function of this group of quartic functions is the most basic function in the set: y = x 3 b) Similarities include: end behavior, domain and range, direction, and width c) Differences include: x and y intercepts, increasing and decreasing intervals 4) This set of graphs can be seen at: Graph for Q#4 a) The parent function of this group of quartic functions is the same as problem #3: y = x 3 b) Similarities include: x and y intercepts, degree, domain, and direction c) Differences include: width and increasing/decreasing intervals Practice 1. Explain what a Square Function is: 2. What is a cube function? 3. Describe the rate of growth of a cubic function related to the growth of a squared function 4. For square root functions we have to define the function value as the positive root, also known as what? 5. Why are reciprocal functions asymptotic to the x-axis? Identify the parent function within each set of squared functions. Graph each set of functions using a graphing calculator. Identify similarities and differences of each set. Set 1: TABLE 1.29: f (x) = x 2 10 f (x) = x 2 1 f (x) = x 2 f (x) = x f (x) = x

74 1.11. Square and Cube Function Families 6. Set 1: a) Parent Function: b) Similarities: c) Differences: Set 2: TABLE 1.30: f (x) = (x + 10) 2 f (x) = (x + 4) 2 f (x) = x 2 f (x) = (x 2) 2 f (x) = (x 5) 2 7. Set 2: a) Parent Function: b) Similarities: c) Differences: Set 3: TABLE 1.31: f (x) = x 2 f (x) = x 2 8. Set 3: a) Parent Function: b) Similarities: c) Differences: Use the above information and the vertex form of a quadratic equation: f (x) = a(x h) 2 +kto help you answer the following questions: 9. How does the a value affect the graph? 10. How does the h value affect the graph? 11. How does the k value affect the graph? 12. How are domain values similar/different? 13. How are range values similar/different? 14. Does the a, h, and/or k value affect the domain? 15. Does the a, h, and/or k value affect the range? Cubic Functions: Circle the parent function within each set of cubic functions. Graph each set of functions using a graphing calculator. Identify similarities and differences of each set. Set 4: TABLE 1.32: f (x) = x 3 5 f (x) = x 3 3 f (x) = x 3 f (x) = x f (x) = x Set 4: a) Parent Function: b) Similarities: c) Differences: Set 5: TABLE 1.33: f (x) = x 3 f (x) = x Set 5: a) Parent Function: b) Similarities: c) Differences: 70

75 Chapter 1. Analyzing Functions 1.12 Vertical and Horizontal Transformations Here you will learn about graphing more complex types of functions easily by applying horizontal and vertical shifts to the graphs of parent functions. If you are not familiar with parent functions or function families, it would be a good idea to review the lessons on those topics before proceeding. Horizontal and vertical transformations are two of the many ways to convert the basic parent functions in a function family into their more complex counterparts. What vertical and/or horizontal shifts must be applied to the parent function of y = x 2 in order to graph g(x) = (x 3) 2 + 4? Watch This Embedded Video: MEDIA Click image to the left for more content. James Sousa - Function Transformations: Horizontal and Vertical Translations Guidance Have you ever tried to draw a picture of a rabbit, or cat, or dog? Unless you are talented, even the most common animals can be a bit of a challenge to draw accurately (or even recognizably!). One trick that can help even the most "artistically challenged" to create a clearly recognizable basic sketch is demonstrated in nearly all "learn to draw" courses: start with basic shapes. By starting your sketch with simple circles, ellipses, rectangles, etc., the basic outline of the more complex figure is easily arrived at, then details can be added as necessary, but the figure is already recognizable for what it is. The same trick works when graphing equations. By learning the basic shapes of different types of function graphs, and then adjusting the graphs with different types of transformations, even complex graphs can be sketched rather easily. This lesson will focus on two particular types of transformations: vertical shifts and horizontal shifts. We can express the application of vertical shifts this way: Formally: For any function f (x), the function g(x) = f (x) + c has a graph that is the same as f (x), shifted c units vertically. If c is positive, the graph is shifted up. If c is negative, the graph is shifted down. Informally: Adding a positive number after the x outside the parenthesis shifts the graph up, adding a negative (or subtracting) shifts the graph down. We can express the application of horizontal shifts this way: Formally: given a function f (x), and a constant a >0, the function g(x) = f (x - a) represents a horizontal shift a units to the right from f (x). The function h(x) = f (x + a) represents a horizontal shift a units to the left. 71

76 1.12. Vertical and Horizontal Transformations Informally: Adding a positive number after the x inside the parenthesis shifts the graph left, adding a negative (or subtracting) shifts the graph right. Example A What must be done to the graph of y = x 2 to convert it into the graphs of y = x 2-3, and y = x 2 + 4? Solution: At first glance, it may seem that the graphs have different widths. For example, it might look like y = x 2 + 4, the uppermost of the three parabolas, is thinner than the other two parabolas. However, this is not the case. The parabolas are congruent. If we shifted the graph of y = x 2 up four units, we would have the exact same graph as y = x If we shifted y = x 2 down three units, we would have the graph of y = x 2-3. Example B Identify the transformation(s) involved in converting the graph of f (x) = x into g(x) = x - 3. Solution: From the examples of vertical shifts above, you might think that the graph of g(x) is the graph of f(x), shifted 3 units to the left. However, this is not the case. The graph of g(x) is the graph of f (x), shifted 3 units to the right. The direction of the shift makes sense if we look at specific function values. TABLE 1.34: x g(x) = abs(x - 3)

77 Chapter 1. Analyzing Functions TABLE 1.34: (continued) x g(x) = abs(x - 3) From the table we can see that the vertex of the graph is the point (3, 0). The function values on either side of x = 3 are symmetric, and greater than 0. Example C What transformations must be applied to y = x 2, in order to graph g(x) = (x + 2) 2 2? Solution The graph of g(x) = (x + 2) 2 2 is the graph of y = x 2 shifted 2 units to the left, and 2 units down. TABLE 1.35: Were you able to solve the question at the beginning of the lesson? "What transformations must be applied to y = x 2, in order to graph g(x) = (x 3) 2 + 4?" The graph of g(x) = (x 3) is the graph of y = x 2 shifted 3 units to the right, and 4 units up. If you were able to identify the translation before the review, congratulations! You are on your way to an excellent conceptual base for manipulating functions. 73

78 1.12. Vertical and Horizontal Transformations Vocabulary A shift, also known as a translation or a slide, is a transformation applied to the graph of a function which does not change the shape of the graph, only the location. Vertical shifts are a result of adding a constant term to the value of a function. A positive term results in an upward shift, and a negative term in a downward shift. Horizontal shifts are produced by adding a constant term to the function inside the parenthesis. A positive term results in a shift to the left and a negative term in a shift to the right (easily confused, pay attention!). Guided Practice Questions: 1) Use the graph of y = x 2 to graph the function y = x ) What is the relationship between f (x) = x 2 and g(x) = (x - 2) 2? 3) What is the relationship between f(x) = x 2-6 and f(x) = x 2? 4) Use the parent function f(x) = x 2 to graph f(x) = x ) Use the parent function f(x) = x to graph f(x) = x - 4. Solutions: 1) The graph of y = x 2 is a parabola with vertex at (0, 0). The graph of y = x 2-5 is therefore a parabola with vertex (0, -5). To quickly sketch y = x 2-5, you can sketch several points on y = x 2, and then shift them down 5 units. 2) The graph of g(x) is the graph of f (x), shifted 2 units to the right. 3) Adding or subtracting a value outside the parenthesis results in a vertical shift. Therefore, the graph of f(x) = x 2-6 is the same as f(x) = x 2 shifted 6 units down. 74

79 Chapter 1. Analyzing Functions 4) The function f(x) = x 2 is a parabola with the vertex at (0, 0). As we saw in Q #3, adding outside the parenthesis shifts the graph vertically. Therefore, f(x) = x will be a parabola with the vertex 3 units up. 5) The graph of the absolute value function family parent function f(x) = x is a large "V" with the vertex at the origin. Adding or subtracting inside the parenthesis results in horizontal movement. Recall that the horizontal shift is right for negative numbers, and left for positive numbers. Therefore f(x) = x - 4 is a large "V" with the vertex 4 units to the right of the origin. 75

80 1.12. Vertical and Horizontal Transformations Practice 1. Graph the function f (x) = 2 x 1 3 without a calculator. 2. What is the vertex of the graph and how do you know? 3. Does it open up or down and how do you know? 4. For the function: f (x) = x +c if c is positive, the graph shifts in what direction? 5. For the function: f (x) = x +c if c is negative, the graph shifts in what direction? 6. The function g(x) = x a represents a shift to the right or the left? 7. The function h(x) = x + a represents a shift to the right or the left? 8. If a graph is in the form a f (x). What is the effect of changing the a? Describe the transformation that has taken place for the parent function f (x) = x 9. f (x) = x f (x) = 5 x + 7 Write an equation that reflects the transformation that has taken place for the parent function g(x) = 1 x, for it to move in the following ways: 11. Move two spaces up 12. Move four spaces to the right 13. Stretch it by 2 in the y-direction Write an Equation for each described transformation. 14. a V-shape shifted down 4 units. 15. a V-shape shifted left 6 units 16. a V-shape shifted right 2 units and up 1 unit. The following graphs are transformations of the parent function f (x) = x in the form of f (x) = a x h = k. Graph or sketch each to observe the type of transformation. 76

81 Chapter 1. Analyzing Functions 17. f (x) = x +2. What happens to the graph when you add a number to the function? (i.e. f(x) + k). 18. f (x) = x 4. What happens to the graph when you subtract a number from the function? (i.e. f(x) - k). 19. f (x) = x 4. What happens to the graph when you subtract a number in the function? (i.e. f(x - h)). 20. f (x) = x + 2. What happens to the graph when you add a number in the function? (i.e. f(x + h)). Practice: Graph the following: 21. f (x) = 2 x 22. f (x) = 5 2 x 23. f (x) = 1 2 x 24. f (x) = 2 5 x 25. Let f (x) = x 2. Let g(x) be the function obtained by shifting the graph of f (x) two units to the right and then up three units. Find a formula forg(x) and then draw its graph Suppose H(t) gives the height of high tide in Hawaii(H) on a Tuesday, (t) of the year. Use shifts of the function H(t) to find formulas of each of the following functions: 26. F(t), the height of high tide on Fiji on Tuesday (t), given that high tide in Fiji is always one foot higher than high tide in Hawaii. 27. S(d), the height of high tide in Saint Thomas on Tuesday (t), given that high tide in Saint Thomas is the same height as the previous day s height in Hawaii. 77

82 1.13. Stretching and Reflecting Transformations Stretching and Reflecting Transformations Here you will explore how to take the parent function in a function family and either stretch it horizontally or vertically, or reflect it to change its direction. If you are not already familiar with function families, it would be a good idea to review them first, and then return here to apply those concepts. Understanding how changes in the equation of a function result in stretching and/or reflecting the graph of the function is a great way to take some of the mystery out of graphing more complicated equations. By recognizing the family to which a more complex equation belongs, and then identifying what changes have been made to the parent of that family, the graph of even quite detailed functions can be made much more understandable. See if you can identify what parts of the equation: y = 1/5x 2 represent either a stretch or a reflection of the parent function: y = x 2 before the review at the end of this lesson. Watch This Embedded Video: MEDIA Click image to the left for more content. James Sousa - Function Transformations: Horizontal and Vertical Stretches and Compressions Guidance Stretching and compressing graphs If we multiply a function by a coefficient, the graph of the function will be stretched or compressed. Given a function f(x), we can formalize compressing and stretching the graph of f(x) as follows: A function g(x) represents a vertical stretch of f (x) if g(x) = cf (x) and c >1. A function g(x) represents a vertical compression of f (x) if g(x) = cf (x) and 0 <c <1. A function h(x) represents a horizontal compression of f (x) if h(x) = f (cx) and c >1. A function h(x) represents a horizontal stretch of f (x) if h(x) = f (cx) 0 <c <1. Notice that a vertical compression or a horizontal stretch occurs when the coefficient is a number between 0 and 1. Reflecting graphs over the y-axis and x-axis Consider the graphs of the functions y = x 2 and y = -x 2, shown below. 78

83 Chapter 1. Analyzing Functions The graph of y = -x 2 represents a reflection of y = x 2, over the x-axis. That is, every function value of y = -x 2 is the negative of a function value of y = x 2. In general, g(x) = -f (x) has a graph that is the graph of f (x), reflected over the x-axis. Example A Identify the graph of the function y = (3x) 2. Solution We have multiplied x by 3. This should affect the graph horizontally. However, if we simplify the equation, we get y = 9x 2. Therefore the graph if this parabola will be taller/thinner than y = x 2. Multiplying x by a number greater than 1 creates a horizontal compression, which looks like a vertical stretch. Example B Identify the transformation described by y = ((1/2)x) 2. Solution If we simplify this equation, we get y = (1/4) x 2. Therefore multiplying x by a number between 0 and 1 creates a horizontal stretch, which looks like a vertical compression. That is, the parabola will be shorter/wider. Example C Sketch a graph of y = x 3 and y = -x 3 on the same axes. Solution: 79

84 1.13. Stretching and Reflecting Transformations At first the two functions might look like two parabolas. If you graph by hand, or if you set your calculator to sequential mode (and not simultaneous), you can see that the graph of y = -x 3 is in fact a reflection of y = x 3 over the x-axis. However, if you look at the graph, you can see that it is a reflection over the y-axis as well. This is the case because in order to obtain a reflection over the y-axis, we negate x. In other words, h(x) = f (-x) is a reflection of f (x) over the y-axis. For the function y = x 3, h(x) = (-x) 3 = (-x) (-x) (-x) = -x 3. This is the same function as the one we have already graphed. It is important to note that this is a special case. The graph of y = x 2 is also a special case. If we want to reflect y = x 2 over the y-axis, we will just get the same graph! This can be explained algebraically: y = (-x) 2 = (-x) (-x) = x 2. Concept question wrap-up: TABLE 1.36: Are you able to identify the transformations described in the beginning of the lesson now? The function: y = 1/5x 2 is the result of transforming y = x 2 by: reflecting it over the x axis, because of the negative co-efficient on the x. and: vertically compressing it (making it wider), because of the co-efficient being a fraction between 0 and 1. Vocabulary Reflections are transformations which result in a "mirror image" of a parent function. They are a result of differing signs between parent and child functions. 80

85 Chapter 1. Analyzing Functions Stretches are transformations which result in the width of a graph being increased or decreased. They are the result of the co-efficient of the x term being between 0 and 1. Guided Practice Questions 1) Graph the functions y = x and y = x. 2) Sketch the graph of y = 3x 2 by appropriately stretching the parent graph y = x 2 3) Sketch the graph of y = 3x 2 by reflecting the graph of y = 3x 2 above. 4) Sketch the graph of y = x by appropriately stretching y = 3x 5) Identify the function and sketch the graph of y = x reflected over both axes Answers 1) The equation y = x might look confusing because of the - x under the square root. It is important to keep in mind that - x means the opposite of x. Therefore the domain of this function is restricted to values 0. For example, if x = - 4, y = ( 4) = 4 = 2. It is this domain, which includes all real numbers not in the domain of y = x plus zero, that gives us a graph that is a reflection over the y-axis. In sum, a graph represents a reflection over the x-axis if the function has been negated (i.e. the y has been negated if we think of y = f (x)). The graph represents a reflection over the y-axis if the variable x has been negated. 2) The graph of y = 3x 2 is the graph of the parent, y = x 2, with each y-coordinate multiplied by 3. The image below shows both the parent and the child function on the same axes. 81

86 1.13. Stretching and Reflecting Transformations 3) The graph of y = 3x 2 is the graph of y = 3x 2 reflected over the x-axis, the image below shows both functions. 82 4) The graph of y = 3x is the graph of y = x with each co-ordinate multiplied by 3, the image below shows both graphs.

87 Chapter 1. Analyzing Functions 5) To reflect the graph of y = x over both axes, the function must be negated both outside and inside the root: y = x. The negation (negative) outside of the root has the effect of reflecting the graph vertically, and the negation inside of the root reflects the graph horizontally. The image below shows three versions: a) (BLUE) y = x b) (GREEN) y = x c) (RED) y = x Practice 1. If a function is multiplied by a coefficient, what will happen to the graph of the function? 2. What does multiplying x by a number greater than one create? 3. What happens when we multiply x by a number between 0 and 1 4. In order to obtain a reflection over the y axis what do we have to do to x? 5. how do we obtain a reflection over the x-axis? 6. Write a function that will create a horizontal compression of the following: f (x) = x Write a function that will horizontally stretch the following: f (x) = x Rewrite this function f (x) = xto get a reflection over the x-axis. 9. Rewrite this function f (x) = xto get a reflection over the y-axis. Graph each of the following using transformations. Identify the translations and reflections. 10. f (x) = x 2 83

88 1.13. Stretching and Reflecting Transformations h(x) = x g(x) = 1 x f (x) = 4x h(x) = (x + 3) f (x) = 1 3 (x 3) f (x) = 4 x f (x) = 2 3(x 2) Let y = f (x) be the function defined by the line segment connecting the points (-1, 4) and (2, 5). Graph each of the following transformations of y = f (x). 18. y = f (x) y = f (x + 2) 20. y = f ( x) 21. y = f (x + 3) 2 The graph of y = x is shown below. Sketch the graph of each of the following transformations of y = x 22. y = x y = x y = x 84

89 Chapter 1. Analyzing Functions 1.14 Combining Transformations Here you will learn about combining different types of transformations applied to a single graph. How do the different forms of transformations result in the differences between the basic parent functions we have explored and some of the more complex graphs you may have seen? It has likely occurred to you that these individual transformations are not enough to result in such significant differences. How do we then apply the individual transformations so that the more complex graphs may be understood? Watch This Embedded Video: MEDIA Click image to the left for more content. James Sousa - Graphing Multiple Function Transformations - Part 1 of2 Guidance Combining transformations By combining shifts, reflections, and vertical and horizontal stretches and compressions, a simple parent function graph can represent a much more advanced function. Consider the equation y = 2(x - 3) We can compare the graph of this function to the graph of the parent y = x 2 : the graph represents a vertical stretch by a factor of 2, a horizontal shift 3 units to the right, and a vertical shift of 1 unit. We can use this relationship to graph the function y = 2(x - 3) You can start by sketching y = x 2 or y = 2x 2. Then you can shift the graph 3 units to the right, and up 1 unit. 85

90 1.14. Combining Transformations Example A Graph the function below using your knowledge of the parent function y = x and your knowledge of transformations. f (x) = - x + 3 Solution: f (x) = - x + 3 The parent graph of this function is the graph of y = x, reflected over the x-axis, and shifted up 3 units. The question is: which transformation do you perform first? We can answer this question if we consider a few key function values. The table below shows several function values for f (x) = - x + 3: TABLE 1.37: x f (x) = - abs(x) abs(-3) + 3 = -(+3) + 3 = = 0-2 -abs(-2) + 3 = -(+2) + 3 = = From the function values in the table we can see that the function increases until a vertex at (0, 3), and then it decreases again. This tells us that we can obtain the graph if we first reflect y = x over the x-axis (turn the v upside down), and then shift the graph up 3 units. We can also justify this ordering of the transformations of we think about the order of operations. To find any function value we take an x value, find its absolute value, find the negative of that number, and then add 3. This is the same as the order of the transformation: reflection comes before shifting up. 86

91 Chapter 1. Analyzing Functions Example B Describe the relationship between the graphs of f (x) = 3(x + 7) and g(x) = x 3. Solution The graph of f (x) = 4(x + 8) 3 3 is the graph of g(x) = x 3, stretched vertically (made narrower) by a factor of 3, shifted 7 units to the left, and then shifted 5 units up. Example C Graph the function below using your knowledge of the parent function y = x and your knowledge of transformations. g(x) = -x + 3 Solution: This function represents a horizontal shift of y = x, and a reflection over the x-axis. Before graphing, consider a few function values: TABLE 1.38: x g(x) = abs( x + 3) abs(-(-3) + 3) = abs(3 + 3) = abs(6) = 6 abs(-(-2) + 3 ) = (2 + 3) = abs(5) = From the values in the table, we can see that the vertex of the graph is at (3, 0). The graph is shown below. The graph looks the same as the graph of y = x - 3. This is the case because y = -x + 3 = -(x - 3), and because - a 87

92 1.14. Combining Transformations = a for all values of a, then -(x - 3) = x - 3. So the original function is equal to x - 3. We can still think of this graph as a reflection: if we reflect y = x over the x-axis, the graph remains the same, as it is symmetric over the x-axis. Then we shift the graph 3 units to the right. What is important to note here is that in order to read the equation as a horizontal shift, the entire expression inside the function (in this case, inside the absolute value) must be negated. Vocabulary A shift is a transformation of a function that does not change the shape of the graph, only the position. A stretch or compression is a function transformation that makes a graph narrower or wider, without changing its position. A reflection is a function transformation that mirrors the graph vertically or horizontally. Guided Practice 1) Sketch the graph of y = 3(x + 2) ) Sketch the graph of y = 1 x ) Sketch the graph of f (x) = 2(x 1) 2 4) Sketch the graph of f (x) = 2 x 1 Answers Remember that the key to multiple transformations is to do them in order. Another way to keep track of which operations to do in which order is to just do them in the order they appear in the equation, left to right. 1) To sketch y = 3(x + 2) 2 + 4, we start with the parent function, y = x a) First we reflect the function over the x axis:

93 Chapter 1. Analyzing Functions b) Next we stretch by 3: c) Shift left by 2: 89

94 1.14. Combining Transformations d) Finally shift up by 4: 2) To sketch y = 1 x start with the parent equation: f (x) = x and complete the transformations left to right: 90

95 Chapter 1. Analyzing Functions a) First reflect over the x axis: b) Second, shift left by 2: c) Finally, shift down by 3: 3) To sketch f (x) = 2(x 1) 2 start with the parent f (x) = x 2 91

96 1.14. Combining Transformations a) First stretch by 2: b) Second, shift right by 1: 4) To sketch the graph of f (x) = 2 x 1 start with the parent y = x 92

97 Chapter 1. Analyzing Functions a) First reflect over the x axis: b) Second, stretch by 2: c) Third, shift right by 1: Practice 1. What part of the function g(x) = ( f (x) + 1) = (x 3 + 1) shifts the graph of f (x) vertically? 2. What part of the function g(x) = ( f (x) + 1) reflects the graph of f (x) across the x-axis? 3. What is different between the functions g(x) = (x ) and h(x) = x that changes the appearance of the graph? This example shows the effect of shifting a graph vertically and then stretching it vertically. 93

98 1.14. Combining Transformations 4. Given the functiong(x) = 3.0( f (x) + 2.0) = 3.0(x ), what is it that shifts the graph of f (x) vertically? a) 3.0 b)x 2 c) Given the functiong(x) = 3.0( f (x) + 2.0) = 3.0(x ), what is it that stretches the graph of f (x) vertically? a) 3.0 b)x 2 c) What part of the equation k(x) = (x + 1) 3 shifts the graph of j(x) = x 3 horizontally? 7. What part of the equationk(x) = (x + 1) 3 reflects the graph of j(x) = x 3 across the x-axis? 8. Given the functiong(x) = 3.0( f (x)+2.0) = 3.0(x +2.0) 3, what is it that shifts the graph of f (x) horizontally? 9. Given the functiong(x) = 3.0( f (x)+2.0) = 3.0(x ) 3, what is it that stretches the graph of f (x) vertically? 10. The graph of g(x) is f (x) reflected across the x-axis. The graph of h(x) is f (x) reflected across the y-axis. The graph of j(x) is f (x) reflected across the x-axis and the y-axis. When graphing j(x) does the order in which the reflections occur matter? (Does it matter which axis we reflect the graph across first?) 11. Given the function f (x) = x 3, write a function g(x) that is: f (x) reflected across the y-axis and then stretched vertically by How do you transform the graph of: f (x) = x 3 so that it looks like the graph of: f (x) = 4x 3 + 6? a)stretch it by a factor of 1 4 and shift it up 6 units. b)stretch it by a factor of 6 and shift it left 4 units. c)stretch it by a factor of 4 and shift it down 6 units. d)stretch it by a factor of 4 and shift it up 6 units. 13. How do you transform the graph of: f (x) = x so that it looks like the graph of: f (x) = x 4? a)reflect it across the x-axis and shift it 4 units down. b)reflect it across the y-axis and shift it 4 units up. c)reflect it across the x-axis and shift it 4 units up. d)reflect it across the y-axis and shift it 4 units down. 14. The graph below is a transformation of a common function. What is the common function that has been transformed? a)y = x b)y = x c)y = x 2 d)y = x How has the function in Q# 14 been transformed? a) Reflected across the y-axis and shifted right 3 units. b) Stretched vertically by a factor of 3 and shifted right 1 unit. c) Reflected across the x-axis and shifted up 3 units. d)reflected across the x-axis and shifted left 4 units. 16. Write a function g(x) whose graph looks like the graph of f (x) = x reflected across the x-axis and shifted up 1 unit. g(x) = 17. Choose the function whose graph looks like the graph of f (x) = x 3 shifted to the right 2 units and reflected across the y-axis. a) f (x) = ( x 2) 3 b) f (x) = ( x + 2) 3 c) f (x) = (x 2) 3 d) f (x) = (x 2) 3 94

99 Chapter 1. Analyzing Functions 1.15 Operations on Functions Here you will learn how to perform the standard mathematical operations of addition, subtraction, multiplication, and division on functions. You will also explore the graphs that result from these operations. Just as numbers can be added, subtracted, multiplied, and divided, so too can functions. Combining functions in this way can often have surprising results, as the resultant function may not have a graph that appears similar to that of either input function s graph. How can you tell, before completing the entire operation and graphing the result, whether the new function is likely to resemble one of the input functions? How do you describe combined functions without a graph? Watch This Embedded Video: MEDIA Click image to the left for more content. James Sousa - The Algebra of Functions Guidance Sums and Differences of Functions Consider the function: f (x) = 48/x + 2x 2. Notice that the equation has two terms: The first term: 48/x The second term: 2x 2 Therefore we can think of the function f (x) as the sum of two other functions: The reciprocal function g(x) = 48/x The quadratic function b(x) = 2x 2 When we add the functions together, we get a new type of graph that resembles both the graphs of g(x) and b(x): 95

100 1.15. Operations on Functions The graph on the right is f (x). The right portion of f (x) resembles the parabola b(x), but is asymptotic to the y-axis. The left portion of f (x) resembles the left side of g(x), as both functions are asymptotic to the negative y-axis. There are two points to be stressed here: first, that we can add functions together, and second, that the resulting sum may be a different kind of function from the original two. The sum or difference of a function is more likely to resemble the original two functions if they are from the same family. For example, if two functions from the linear function family are added together, the sum function is also a member of the linear family. Example A If f (x) = x 3 + 2x 2 and g(x) = x 2-5, what is f - g? What does the graph look like? Solution: The difference is: f - g = x 3 + 2x 2 - (x 2-5) = x 3 + 2x 2 - x = x 3 + x The graph of the new function, along with f (x), is shown here: Because f (x) and the new function y = x 3 + x are both members of the cubic family, they have similar shapes. 96

101 Chapter 1. Analyzing Functions To recap: When we add or subtract functions, the resulting sum or difference function may be in the same family as one or both of the original functions, or it may be a different type of function. The resultant function is more likely to be in the same family if both of the initial functions are in the same family as each other. Example B Given f (x) = 2x 2 and g(x) = x + 1, find r(x) = f (x)/ g(x) and t(x) = g(x)/f (x). Solution: r(x) = f (x)/g(x) = 2x 2 /(x + 1). This is a rational function, and does not have a horizontal asymptote. It does, however, have a vertical asymptote at x = -1, as the domain excludes x = -1. t(x) = g(x)/ f (x) = (x + 1)/2x 2. This is also a rational function. This function has a horizontal asymptote at y = 0 (the x-axis), and a vertical asymptote at x = 0 (the y-axis). Notice that the graph of this function crosses its asymptote at (-1, 0), but then as x approaches, the function values approach 0. 97

102 1.15. Operations on Functions In general, if we multiply linear and polynomial functions (quadratics, cubics, and other such functions with higher exponents, such as y = x 4 + 3x 2 + 2), we will obtain other polynomial functions. If we divide these kinds of functions, we will obtain other polynomial functions, or rational functions. Multiplying and dividing other types of functions may result in more complicated graphs. Example C Consider the functions f (x) = x and g(x) = x 2-4. Identify the graphs of f (x)/g(x) = x /(x 2-4) and g(x)/f (x) = (x 2-4)/ x. Solution: The graphs of these two functions are not unlike the rational functions discussed in a later lesson. TABLE 1.39: Did you discover the trick for identifying when a resultant function graph is likely to resemble the input graphs, as mentioned at the beginning of the lesson? The sum or difference of a function is more likely to resemble the original two functions if they are from the same family. In other words, if you are adding or subtracting two quadratic equations, the result is likely to be quadratic, and have a similar graph. Vocabulary Function sum: The result of the addition of two functions. Function difference: The result of the subtraction of two functions. Asymptote: A line on a graph toward which the output of a given function may approach, but never quite reach. Guided Practice Questions 1) Given f (x) = 4x 2 7 and g(x) = 3x 2 2x + 8: Find and graph (use technology): ( f + g)(x) 2) Multiply the function by the scalar value If f (x) = 3x + 10 find 3 f (x) 98

103 Chapter 1. Analyzing Functions 3) Given f (x) = 3x 7 and g(x) = 4x + 6: Solutions Find and graph (use technology) ( f g)(x) 1) Step 1: Recall that ( f + g)(x) = f (x) + g(x) Step 2: Substitute f (x) + g(x) = (4x 2 7) + (3x 2 2x + 8) Step 3: Combine like terms ( f + g)(x) = 7x 2 2x + 1 So our answer is: ( f + g)(x) = 7x 2 2x + 1 The graph of f (x) = 7x 2 2x + 1 looks like: 2) To multiply a function by a scalar, multiply each term of the function by the scalar: Step 1: Substitute: 3 f (x) = 3(3x + 10) Step 2: Distribute: 3 f (x) = 9x + 30 So our answer is: 3 f (x) = 9x ) Step 1: Recall that ( f g)(x) = f (x) g(x) Step 2: Substitute: f (x) g(x) = (3x 7)(4x + 6) Step 3: Distribute (FOIL): ( f g)(x) = 12x x 28x 42 Step 4: Combine like terms: ( f g)(x) = 12x 2 10x 42 99

104 1.15. Operations on Functions So our answer is:( f g)(x) = 12x 2 10x 42 The graph of f (x) = 12x 2 10x 42 looks like this: Practice Given f (x) = x3 x+1 and g(x) = x(x + 1) find each of the following: 1. ( f g)(x) = 2. ( f g)( 1) = 3. ( f g )(x) = Simplify the following: 4. If f (x) = 2x + 4and g(x) = 3x 7, find ( f + g(x)). 5. If g(x) = 2 3 x + 12 and h(x) = 1 4x + 7, find(g + h)(x) 6. If f (x) = 4x 2 10 and g(x) = 5x 2 2x 3, find( f + g)(x) 7. If f (x) = 6x 2 3x + 5 and g(x) = 4x 2 + 5x 8, find(g f )(x). 8. Ifg(x) = 6x 8, find 3 2 g(x). 9. If g(x) = 2x and h(x) = 3x 6, find(g h)(x). Evaluate and Graph: 10. if f (x) = 6x + 4 and g(x) = 7x 8, find( f + g)(3). 11. If f (x) 1 4 x + 3 and h(x) = 3 2x + 6, find (g + h)(12). 100

105 Chapter 1. Analyzing Functions 12. Ifg(x) = 5x 2 4x + 3andh(x) = 2x 7, find(g h)(2). 13. Ifg(x) = 4x 3 3x find 5g(6). 14. Ifh(x) = 4x 7 find 2h( 5). 15. If f (x) = x + 4 and g(x) = 3x 6, find ( f g)(1). 16. Ifh(x) = x 4 andg(x) = x 12, find(h g)( 2) Try these more challenging problems. Solve and graph. 17. If f (x) = 4x 7, g(x) = 3x + 18, and h(x) = 5x + 2, find( f + g h)(x). 18. If f (x) = 6x 8,y(x) 1 2x, and h(x) = x + 4, find ( f g h)(x) 19. If g(x) = 3x 7 and (g h)(x) = 15x 2 47x + 28, find(h)(x). 101

106 1.16. Composition of Functions Composition of Functions Here you will explore the composition of functions. Function composition refers to the combination of two functions by using the output of one function as the input of the other. If f(x) = x + 2, and g(x) = 2x + 4, what is f(g(x))? A function can be conceptualized as a black box. The input, or x value is placed into the box, and the box performs a specific set of operations on it. Once the operations are complete, the output (the "f(x)" or "y" value) is retrieved. Once the output is retrieved, the box is ready to work on the next input. Using this idea, function composition can be seen as a box inside of a box. The input x value goes into the inner box, and then the output of the inner box is used as the input of the outer box. This lesson is all about boxes inside of boxes. See if you can use what you learn to answer the question above before the review at the end. Watch This Embedded Video: MEDIA Click image to the left for more content. James Sousa - Composite Functions Guidance Composition of Functions Functions are often described in terms of input and output. For example, consider the function f (x) = 2x + 3. When we input an x value, we output a y value, or a function value. We find the output by taking the input x, multiplying by 2, and adding 3. We can do this for any value of x. Now consider a second function g(x) = 5x. For this function too, we can take an x value, input the x into g(x), and obtain an output. What happens if we take the output of g and use it as the input of f? Example A Given the function definition above, g(x) = 5x. Therefore if x = 4, then we have g(4) = 5(4) = 20. What happens if we then take the output of 20 and use it as the input of f? Solution: Substituting 20 in for x in f (x) = 2x + 3 gives: f (20) = 2(20) + 3 = 43. The table below shows several examples of this same process: 102

107 Chapter 1. Analyzing Functions TABLE 1.40: x Output from g Output from f Examining the values in the table, we can see a pattern: all of the final output values from f are 3 more than 10 times the initial input. We have created a new function called h(x) out of f (x) = 2x + 3 in which g(x) = 5x is the input: h(x) = f (5x) = 2(5x) + 3 = 10x + 3 When we input one function into another, we call this the composition of the two functions. Formally, we write the composed function as f (g(x)) = 10x + 3 or write it as (f o g)x = 10x + 3 Example B Find f (g(x)) and g(f (x)): TABLE 1.41: a. f (x) = 3x + 1 and g(x) = x 2 b. f (x) = 2x + 4 and g(x) = (1/2)x- 2 Solution: a. f (x) = 3x + 1 and g(x) = x 2 f (g(x)) = f (x 2 ) = 3(x 2 ) + 1 = 3x g(f (x)) = g(3x + 1) = (3x + 1) 2 = 9x 2 + 6x + 1 In both cases, the resulting function is quadratic. b. f (x) = 2x + 4 and g(x) = (1/2) x - 2 f (g(x)) = 2((1/2)x - 2) + 4 = (2/2)x = (2/2)x = x g(f (x)) = g(2x + 4) = (1/2)(2x + 4) - 2 = x+ 2-2 = x. In this case, the composites were equal to each other, and they both equal x, the original input into the function. This means that there is a special relationship between these two functions. We will examine this relationship in Chapter 3. It is important to note, however, that f (g(x) is not necessarily equal to g(f (x)). Example C Decompose the function f (x) = (3x - 1) 2-5 into a quadratic function g(x) and a linear function h(x). Solution: When we compose functions, we are combining two (or more) functions by inputting the output of one function into another. We can also decompose a function. Consider the function f (x) = (2x + 1) 2. We can decompose this function into an inside and an outside function. For example, we can construct f (x) = (2x+ 1) 2 with a linear function and a quadratic function. If g(x) = x 2 and h(x) = (2x + 1), then f (x) = g(h(x)). The linear function h(x) = (2x + 1) is the inside function, and the quadratic function g(x) = x 2 is the outside function. Let h(x) = 3x - 1 and g(x) = x 2-5. Then f (x) = g(h(x)) because g(h(x)) = g(3x - 1) = (3x - 1)

108 1.16. Composition of Functions The decomposition of a function is not necessarily unique. For example, there are many ways that we could express a linear function as the composition of other linear functions. TABLE 1.42: Can you answer the question at the beginning of the lesson now? If f(x) = x + 2, and g(x) = 2x + 4, what is f(g(x))? f(g(x)) = f(2x + 4) = (2x + 4) + 2 = 2x + 6 Once you get the idea, composite functions aren t as difficult as they look! Vocabulary A composite function is a function formed by using the output of one function as the input of another. The input of a function is the value on which the function is performed (commonly the x value). The output of a function is the result of the operations performed on x (commonly y or f(x)). Guided Practice Questions 1) Given: 2) Given: f (x) = 5x + 3 g(x) = 3x 2 Find: f (g(4)) h(n) = 7n (g(n)) g(t) = t f (x) = 2x + g(x) Find: f (h( 5)) 3) Given: g(x) = 5x 2 h(x) = 5x 2 2x 4(g(x)) Find h(g( 1)) Solutions 1) To find f (g(4)), we need to know what g(4) is, so we know what to substitute into f (x): Substitute 4 for x for the function g(x), giving: Simplify: 3 16 = 48 g(4) = 48 Substitute 48 for the x in the function f (x) giving: 5(48) + 3 Simplify: = 243 f (g(4)) = 243 2) First, let s solve for the value of the inner function, h( 5). Then we ll know what to plug into the outer function. h( 5) = (7)( 5) (g( 5)) 104

109 Chapter 1. Analyzing Functions To solve for the value of h, we need to solve g( 5) g( 5) = ( 5) g( 5) = 5 Now we have: h( 5) = (7)( 5) (4)(5) Simplify to get: h( 5) = 14 Now we know that h( 5) = 14. That tells us that f (h( 5)) is f ( 14) Find f ( 14) = ( 2)( 14) + g( 14) So to solve for the value of f ( 14), we need to solve for the value of g( 14) g( 14) = ( 14) g( 14) = 14 Now we can finish up! f ( 14) = ( 2)( 14) + 14 f ( 14) = 42 3) First, solve for the value of the inner function g( 1) to find what to plug into the outer function h(g( 1)) g( 1) = 5( 1) 2 g( 1) = 5 1 g( 1) = 5 Next, solve for h(g( 1)) which we now know is: h(5) h(5) = 5(5 2 ) + ( 2)(5) 4(g(5)) To solve for the value of h, we need to solve for the value of g(5). g(5) = 5(5 2 ) g(5) = 125 h(5) = 5(5 2 ) + ( 2)(5) + ( 4)(125) Finally: h(5) = 385 Practice For problems 1-3: f (x) = 2x 1 g(x) = 3x h(x) = x Find: f (g( 3)) 2. Find: f (h(7)) 3. Find: h(g( 4)) 4. Find: f (g(h(2))) 105

110 1.16. Composition of Functions Evaluate each composition below: 5. Given: f (x) = 5x + 2 and g(x) = 1 2x + 4 Find f (g(12)) 6. Given: g(x) = 3x + 6 and h(x) = 9x + 3 Find g(h( 1 3 )) 7. Given: f (x) = 1 5 x + 4 and g(x) = 4x2 Find f (g(10)) 8. Given g(x) = 3 x 4 +6 and h(x) = x 3 Find h(g(4)) 9. Given f (x) = x + 2 and g(x) = 2x Find g( f ( 7)) 10. Given f (x) = 3x + 2 and given g(x) = 2x 2 and given h(x) = 4 7 x +6 Find f (g(h(1))) 11. Given f (x) = ( 3) and given g(x) = 2x and given h(x) = 4x 12 Find f (h(g(18))) 12. Are compositions commutative? In other words, does f (g(x)) = g( f (x))? 13. Given: f (x) = 2 2 5x and h(x) = 3x + 2 Find f (h(x)) 14. Two functions are inverses of each other if f (g(x)) = x and g( f (x)) = x If f (x) = x + 3, find its inverse: g(x) 15. A toy manufacturer has a new product to sell. The number of units to be sold, n, is a function of the price p such that: n(p) = 30 25p The revenue r earned from the sales is a function of the number of units sold n such that: r(n) = x2 Find the function for revenue in terms of price, p. 106

111 Chapter 1. Analyzing Functions 1.17 Linear and Quadratic Models Here you will learn about applying what you know of functions to mathematically describe real-life situations and objects. Scientists and social scientists use mathematical models to understand a wide variety of quantifiable phenomena, from the workings of subatomic particles, to how people will function in the economy. Learning to express real-life situations as mathematical functions allows seemingly complex ideas and actions to be broken down into smaller, simpler parts and analyzed. How might you express the following mathematically? Two brothers decide to race home from school, taking different routes. The second brother leaves 5 minutes after the first, and both arrive at home at the same time. See if you can write an appropriate expression before the review at the end of the lesson that explains the process. Watch This Embedded Video: MEDIA Click image to the left for more content. Watch this video on the number of handshakes at a party to learn more about linear and quadratic models. Guidance Linear Models The simplest functions are generally linear models. For instance, the equation y = 3x could be used to represent how much money you would bring in if you sold x boxes of cookies for $3 per box. Many situations can be modeled with linear functions. The key idea is that some quantity in the situation has a constant rate of change. In the cookieselling example, every box costs $3.00. Therefore the profits increase at a constant rate. In sum, linear functions are used to model a situation of constant change, either increase or decrease. Quadratic models Quadratic functions may also be used as models. For example, the function A(x) = 50x x2 2 could be used to represent the area of a rectangular plot of land enclosed on three sides by 100 feet of fence. This model could be used to identify the maximum possible area for the plot of land. Another example is a situation in which a ball is tossed into the air. The ball will travel up, and then it will travel down until it hits the ground. How high will the ball go? When will it reach the ground? This kind of situation can be modeled by a function of the form h(t) = -16t 2 + v 0 t + h 0. The variable t represents the time since the ball was thrown. The coefficient v 0 represents the initial velocity of the ball, and the constant h 0 represents the initial height of the ball. The constant -16 comes from the force of gravity which pulls the ball down, which is why it is negative. Example B shows a specific situation of this form. 107

112 1.17. Linear and Quadratic Models Example A You run a lawn mowing business, and charge $15 per lawn. Write a linear function to describe the amount of money made as a function of the number of lawns mowed. Solution If you express the number of lawns mowed as l, then $15 multiplied by l would represent the total money made based on the number of lawns, therefore: 15(l) = total income. Example B You drop a rock off the edge of a cliff, and time the fall in seconds. Determine the function that models the height of the cliff as a function of falling time in seconds. Solution With an initial velocity of 0, the height of the cliff may be calculated with the formula: h(t) = 16t 2. Example C You are standing on the roof of a building that is 20 feet above the ground. You toss a ball into the air with an initial vertical velocity of 40 ft/sec, so that it will land on the ground, not on the roof. How high will the ball go, and when will it reach its maximum height? When will the ball hit the ground? Solution: First we need to write a function to model the situation. Using the general form of the equation given above, we can write the function h(t) = -16t t + 20, where h(t) represents the height above the ground. To answer the first question, we need to examine the graph of the function. If you graph this equation on your calculator, you will need to determine a good viewing window. One way to start to determine a good window is to take into account the y-intercept of the function. In this case, the y-intercept is (0, 20). Think about what kind of function this is: a parabola, facing downwards. This fact should lead you to think that we need to look at y-values well above 20. It is often useful to look at a table of values. Using the ask capability of the calculator, if you input x values of 1, 2, and 3, you will see that the function goes up to 44 at x = 1. The maximum value is most likely somewhere near x = 1. Press WINDOW, and set xmin = 0, xmax = 3. Then set Ymin = 0 (or a little less, if you want to see the y-axis). Ymax should be no less than 44, though you may want to make it larger, such as 50 or more, just to be sure that you can see the vertex. Once you have set the window, press GRAPH. Now you should see the parabola. To identify the coordinates of the vertex, you can use the MAX function in the CALC menu. Remember that the calculator will ask you to input a left bound, a right bound, and a guess for the maximum. If you use the MAX function, you should find that the coordinates of the vertex are (1.25, 45). This means that 1.25 seconds after the ball is thrown into the air, it reaches a maximum height of 45 feet. 108

113 Chapter 1. Analyzing Functions Now, to answer the second part of the question, we need to determine when the height of the ball is 0. Graphically, we are looking for the x-intercept of the parabola. If you return to the GRAPH screen, you should see that the x-intercept is around 3. If we want to determine the exact value, or at least a good approximation of the x-intercept, we can use the ZERO function. Press 2 nd TRACE to get the CALC menu, and choose option 2, ZERO. Like the MAX function, you need to input a left bound, a right bound, and a guess, although the guess is optional just press ENTER. (Note that the calculator works this way because it is asking you to identify which x-intercept to calculate. The parabola has two x-intercepts, and other functions may have more.) Using the ZERO function, you should find that the x-intercept is approximately This means that the ball reaches the ground in just under 3 seconds. TABLE 1.43: Solution to start-of-lesson question Did you come up with a mathematical model for the question at the start of the lesson? Two brothers decide to race home from school, taking different routes. The second brother leaves 5 minutes after the first, and both arrive at home at the same time. There are a number of different ways to model the information, depending on what part(s) of the information you choose to use. A couple of examples include: If t = the time the second brother took to get home, then t + 5 = the time the first brother took. If t = the time the first brother took to get home, then (t + (t - 5))/2 represents the average time to run home. Your model may be similar, or may be written differently, but should compare different values given in the story problem. Vocabulary A linear model is a function representing a situation involving a constant rate of change. The graph of a linear equation is a straight line. A quadratic model is a function representing a situation with a rate of change that is a squared value. The graph of a quadratic function is a parabola. Guided Practice Questions 1) The Senior class has paid $ to a DJ for the Senior Prom. Tickets for the dance are $15.00 each. a) Express the net income as a function of the number of tickets sold. b) Graph the function and identify any limitations on the domain. 2) The Arlington Freshmen class wants to have a fundraiser. The class wants to buy a number of $4.00 flip-flops and $5.00 baseball hats, and has a total of $100 to spend. a) If f represents the number of flip-flops and b represents the number of baseball hats, write a function to represent the number of flip-flops purchased as a function of unspent monies from baseball hats. b) Using your equation from (a), determine the number of baseball hats that can be bought if 10 flip flops were purchased. 109

114 1.17. Linear and Quadratic Models 3) Studies of the metabolism of alcohol consistently show that blood alcohol content (BAC) declines linearly, after rising rapidly after initial ingestion. In one study, BAC in a fasting person rose to about % after a single drink. After an hour the level had dropped to %. a) Write an equation relating BAC to time in hours after drinking t. b) Assuming that BAC continues to decline linearly (meaning at a constant rate of change), approximately when will BAC drop to 0.002%? Answers 1) To find the net income: a) Let I(n) = net income from n tickets sold. Then the income is found by subtracting the cost of the DJ from the earned money. I(n) = $15 n $1000 b) The domain of the function is limited to positive numbers, since the Senior Class will not sell a negative number of tickets. We can say, therefore, that the domain of the function is n > 0, where n is an integer. 2) To express this information as a function, remember that the question specified that there was $100 to spend, and that any money not spent on hats (at $5 ea) was spent on flip-flops (at $4 ea). a) f = $100 5h 4 b) To calculate how many hats could be bought if 10 pairs of flip flops were purchased, substitute 10 in for f, and solve for h: 10 = 100 5h 4 40 = 100 5h 5h = 60 h = 12 Therefore, if 10 pairs of flip flops were purchased, there would be money left over to buy 12 baseball caps. 3) In order to answer the question, you must express the relationship as an equation and then use the equation. First, define the variables in the function and create a table. The two variables are time and BAC. TABLE 1.44: Time BAC % % Next, calculate the rate of change. TABLE 1.45: Time BAC Rate of change % % (0.008%) This rate of change means when the time increases by 1, the BAC decreases (since the rate of change is negative) by.008. In other words, the BAC is decreasing.008% every hour. Since we are told that BAC declines linearly, we can assume that figure stays constant. 110

115 Chapter 1. Analyzing Functions a) Now we can write an equation with b representing BAC and t the time in hours: b =.008t b) To learn when will the BAC reach.002%, substitute.002 in for b and solve for t..002 =.008t =.008t t = 2 Therefore the BAC will reach.002% after 2 hours. Practice 1. From the number of gas stations in a certain country increased by 100 stations per year. In 2004 there were 1100 gas stations. Write a linear equation for the number of gas stations, (y), as a function of time, (t,) where t = 0 represents the year Find the vertex of the following quadratic functions and then graph them. 2. f (x) = 2x 2 6x f (x) = 3(x + 5) 2 2 At the local downtown 4th of July fireworks celebration, the fireworks are shot by remote control into the air from a pit in the ground that is 12 feet below the earth s surface. 4. Find an equation that models the height of an aerial bomb t seconds after it is shot upwards with an initial velocity of 80 ft/sec. Assume that the bomb decelerates at a rate of 30 ft/sec each second 5. Find the vertex of the quadratic function. 6. What is the maximum height above ground level that the aerial bomb will reach? 7. How many seconds after it was launched will it take to reach that height? 8. A rock is thrown from the top of a 763ft tall building. The distance, in feet, between the rock and the ground t seconds after it is thrown is given by d = 16t 2 2t How long after the rock is thrown is it 430 feet from the ground? 9. Use the vertical motion formula h = 16t 2 +v 0 t +s to find the number of seconds it takes for a rocket launched from the ground with a starting velocity of 96 ft/s to reach an altitude of 45 ft. Round answers to the nearest tenth. 10. The function P = t t models the United States population in millions since Use the function P to predict the year in which the population exceeds 1 billion. 11. For which value of x is f (x) = 10 if f (x) = 4x 2 + 3x? 111

116 1.18. Cubic Models Cubic Models Here you will explore the process of modeling functions with cubed functions or with different functions in different parts of the model (piece-wise functions). Many real-life situations do not easily fit completely within a simple model, so it can be valuable to understand the process of using multiple models in the same situation. How do you write a math problem to represent a physical object? The video below, and the first portion of the lesson thereafter, both describe different challenges involving modeling a box created by cutting corners out of a flat sheet of cardboard and then folding up the sides. In the video, the specific problem is to identify the size of the cardboard needed to result in a given volume. In the lesson, the challenge is to identify the greatest volume possible from a given size sheet of cardboard. How would you mathematically model a divider to split the volume of the box described in the lesson into multiple spaces? Watch This Embedded Video: MEDIA Click image to the left for more content. - James Sousa - Ex: Find the Size of Cardboard Needed to Make a Box with a Given Volume Guidance Cubic functions and piece-wise functions can be used to model real-world situations, allowing you to identify missing bits of information you may need to complete a project. Cubic functions are commonly used to model three-dimensional objects to allow you to identify a missing dimension or explore the result of changes to one or more dimensions. Piece-wise functions may be used to model the interactions of multiple items each previously modeled by a simpler function. Example A Consider a situation similar to the one described in the video above, in which a rectangular piece of cardboard is folded into a box. The folding is made possible by cutting squares out of the four corners of the cardboard. Calculate the maximum volume possible of a box made from a sheet of cardboard 12" x 8". Solution: 112