1 An Introduction to the Rational Function EVALUATED. 4 Draw the graph for the rational function. (x)(y) = 100

Transcription

1 Real Functions Skill Builder 1 An Introduction to the Rational Function EVALUATED CHAPTER 3 1 Complete the table of values given below, using the following rational function rule, y = 24 x x y 4 Draw the graph for the rational function (x)(y) = 18 2 Complete the table of values given below, using the following rational function rule, (x)(y) = 100 x y 5 Given the information below, find the rule that describes the results in the table of values. x y Draw the graph for the rational function y = 16 x Skill Builder: An Introduction to the Rational Function 151

2 Skill Builder 6 Given the information below, find the rule that describes the results in the table of values. x y Given the graph of a rational function below, find the rule of this function. 7 Given the graph of a rational function below, find the rule of this function. 9 Peter wants to build a rectangular work table with an area of 4m 2. If x represents the table s length, and y represents the table s width, what rational function describes the above situation? 10 Using the equation of a rational function, describe the relationship between speed (v) and time (t) for a car traveling a distance of 400km. 152 Chapter 3: Real Functions

3 CRITICAL THINKING AND BIG IDEAS An Introduction to the Rational Functions 1. The different number sets were not all developed at the same time; they came about as mathematicians discovered a need for them. In the beginning there were the Natural Numbers ( ), which allowed people to count: 1, 2, 3, 4... However, they could only add or multiply with this set of numbers. In order to subtract, the Integers ( ) were developed: 3, 2, 1, 0, 1, 2 Now, mathematicians could subtract a larger value from a smaller value and have a way to represent the answer. The rational numbers ( ) came next and each rational number can be represented as p q integers. a. What operation requires the use of rational numbers? b. What is the rule of a rational function? c. How can you identify a rational function from its table of values? where p and q are d. In your own words, define the term asymptote and explain why rational functions contain asymptotes. MHS Critical Thinking and Big Ideas 153

4 2. Create a real world situation that could be modeled with a rational function. Define the situation, state the rule of the rational function, complete the table of values and sketch the graph of the function. x y 154 MHS Critical Thinking and Big Ideas

5 3. Theresa was in a bicycle race at Mont Tremblant last weekend. The race was 80 km on Saturday and 80 km on Sunday. Theresa was feeling much better on Sunday and doubled her average speed from Saturday. Fill in the following table of values, letting x represent Theresa s average speed in kilometres per hour on day 1. Distance (km) Speed (km/ hr) Time (hours) Day 1 x Day 2 If Theresa s total time that she cycled for on the two days was 8 hours, what was her average speed on day 2? MHS Critical Thinking and Big Ideas 155

6 Skill Builder 2 The Basic Quadratic Function (parameter a) EVALUATED 1 Find the rule of the quadratic function, given the table of values below. 3 Find the rule of the quadratic function, given the graph below. x y Find the rule of the quadratic function, given the table of values below. x y Chapter 3: Real Functions

7 Skill Builder 4 Find the rule of the quadratic function, given the graph below. 5 Given the basic quadratic function f(x) = x 2 and the transformed quadratic function g(x) = ax 2 below, in what interval does a belong to? f(x) = x 2 g(x) = ax 2 Skill Builder: The Basic Quadratic Function (parameter a) 157

8 Skill Builder 6 Given the basic quadratic function f(x) = x 2 and the transformed quadratic function g(x) = ax 2 below, in what interval does a belong to? 7 The base of a 90 triangle is four times longer than its height. What is the quadratic function that relates the height to its area? x f(x) = x 2 4x g(x) = ax Chapter 3: Real Functions

9 Skill Builder 8 If x represents the edge length of the cube shown below, what is the quadratic formula that relates the total surface area of the cube to the edge length x? 9 A parabola with a vertex of (0,0) passes through the point (2, 6). What is f (4)? x 10 The vertical distance travelled by a freefalling object as a function of time elapsed is a quadratic function. Given the table of values below, what is the vertical distance travelled by an object after 5 seconds? Time (t) 0s 1s 2s 3s Distance (d) 0m 4.9m 19.6m 44.1m Skill Builder: The Basic Quadratic Function (parameter a) 159

10 CRITICAL THINKING AND BIG IDEAS The Basic Quadratic Function (parameter a) 1. Having a strong understanding of quadratic functions will allow you to solve many problems without having to make a single calculation. Highlight the Axis of Symmetry on the following graph, and then use symmetry to find all of the missing values. (, ) (4.5, 40.5) (-4, 32) (, ) (, ) (2, 8) 2 Substitute the coordinates of a point into the equation y = ax in order to determine the rule of this quadratic function. 160 MHS Critical Thinking and Big Ideas

11 2. Higgins is looking for a new designer rain barrel to catch water from his downspouts, which he will use to water the gardens. The type of rain barrel that he likes comes in various sizes and is priced based on the radius of its base. There are two stores that carry these rain barrels, Lee Valley Tools and Home Depot. The cost of the barrels at Lee Valley Tools is represented by function L(x) and at Home Depot by function H(x). Both functions are in the form f ( x) 2 = ax where: x represents the radius of the base of the barrel in decimetres. L(x) represents the cost of the barrel in dollars at Lee Valley Tools. H(x) represents the cost of the barrel in dollars at Home Depot. Higgins has noticed the following pricing tables in the online catalogues of each store. radius (dm) L(x) $ radius (dm) H(x) $ The model that Higgins is looking for has a radius of 20 decimetres. Where should he make his purchase and how much more expensive would the rain barrel be at the other store? MHS Critical Thinking and Big Ideas 161

12 3. Dexter works for the driver s license bureau and he is part of a team putting forward suggestions for how much a driver s renewal of their license should go up based on how many traffic violations they receive. Dexter is evaluating the following two options. Option A: The cost of the renewal would increase according to the function f v = v, where: ( ) 75 v represents the # of traffic violations the driver committed during the year. f(v) represents the increase in dollars. Option B: The cost of the renewal would increase according to the function 2 g v = 40v, where: ( ) v represents the # of traffic violations the driver committed during the year. g(v) represents the increase in dollars. Dexter has asked you to help him with the evaluation to choose which option should be implemented. Note that while the increases are meant to be punitive toward unsafe drivers, they should not be unfair. Your report should include a table of values for each option as well as a detailed position of which option you suggested and why. 162 MHS Critical Thinking and Big Ideas

13 3 The Basic Exponential Function EVALUATED Skill Builder 1 Is the following exponential function increasing or decreasing? Give reasons to support your answer. ( ) h x 11 = 13 x 4 The graph of an exponential function in the form y = c x passes through the point (3, 512). Find the value of the base c. 2 Is the following exponential function increasing or decreasing? Give reasons to support your answer. y = π x 5 The graph of an exponential function in the form y = c x passes through the point (4, 256). Find the value of the base c. 3 The graph of an exponential function in the form y = c x passes through the point (3, 216). Find the value of the base c. Skill Builder: The Basic Exponential Function 163

14 Skill Builder 6 Sketch the following function by making a table of values (use a calculator if necessary) y = 3 x 8 Find the rule of the exponential function y = c x represented in the following graph (4,1296) 7 Sketch the following function by making a table of values (use a calculator if necessary) x 1 y = Chapter 3: Real Functions

15 Skill Builder 9 Find the rule of the exponential function y = c x represented in the following graph 10 Formulate a conjecture as to how the value of c and the sign of x in the basic exponential function y = c x affect the shape of the curve. (-3,125) -1 1 Skill Builder: The Basic Exponential Function 165

16 CRITICAL THINKING AND BIG IDEAS The Basic Exponential Function 1. Sketch the graph of each of the following functions on the coordinate plane provided. In each case, state the characteristics of the function (increasing or decreasing intervals and the equation of any asymptote) and explain how these can be determined through analyzing the function rule. y = 3 x 1 y = 3 x Increasing / Decreasing interval Increasing / Decreasing interval Asymptote Asymptote How to Analyze the Function Rule: 166 MHS Critical Thinking and Big Ideas

17 2. Each of the following tables of values represent an exponential function in the form x f ( x) = c. For each, determine the rule of the function and fill in the rest of the table of values (show all work). x y 1 4 x y x y Explain the strategy you used to determine the rule and identify a point that cannot be used to find the rule of the function. MHS Critical Thinking and Big Ideas 167

18 3. Samuel is looking at making some investments to save for the future. His investment advisor has suggested a Mutual Fund that grows at an annual interest rate of 8% per year. The function rule representing the growth of the fund is ( ) 1.08 x f x = where x represents the number of years the investment stays in the fund. f(x) represents the growth factor of the fund. a. What would the growth factor of the fund be after two years? b. If Samuel invested $5000 in the fund, how much would his investment be worth after 4 years? c. How long would it take Samuel s investment to double in value? Show all work 168 MHS Critical Thinking and Big Ideas

19 4 The Exponential Function (parameter a) EVALUATED Skill Builder 1 State the y-intercept and explain it s meaning given the function below. f(x) = 4.2(0.5) x x: represents time in years f(x): represents the weight of the substance in grams x 4 1 Sketch the graph of the function y = 20 by making a table of values. 2 2 State the y-intercept and explain it s meaning given the function below. f(x) = 10000(1.08) x x: represents time in years f(x): represents the value in dollars 5 Find the rule of the exponential function in the form y = ac x that is represented in the following table of values. x y Sketch the graph of the function y = 6(3) x by making a table of values. Skill Builder: The Exponential Function (parameter a) 169

20 Skill Builder 6 Find the rule of the exponential function in the form y = ac x that is represented in the following table of values. x y Fish off the coast of South Africa have contracted and are spreading a new type of disease called Blue Gills disease that decreases the amount of oxygen they are able to move through their gills. In 2008 researchers found that there were 6 fish that had the disease and that this amount was doubling at a rate of once per year. Find the function rule in the form y = ac x that is represented in this word problem where x represents time in years and y represents the number of fish with Blue Gills disease. 7 Perdita is an environmental researcher; she is conducting a study on a new type of eco-friendly plastic. She is taking samples that have an initial mass of 324 grams and has concluded that they decompose to half of their mass every month. Find the rule in the form y = ac x that is represented in this word problem where x represents time in months and y represents the size of the sample in grams. 170 Chapter 3: Real Functions

21 Skill Builder 9 Marcus invests $500 in a mutual fund that increases in value according to the function rule A(t) = 500(1.15) x where x represents the time elapsed in years since he made the investment and A(t) represents the value of his investment in dollars. Estimate the amount of time it will take for Marcus investment to double in value. 10 The function R(t) = 16(2) t represents the population of a warren of rabbits where t represents the time in years and R(t) represents the population of rabbits. Estimate the number of years before the population of rabbits reaches 100. Skill Builder: The Exponential Function (parameter a) 171

22 CRITICAL THINKING AND BIG IDEAS The Exponential Function (Parameter a) 1. Trisha is studying for her upcoming quiz on exponential functions in the form y = ac and is having trouble remembering how to determine if the function is increasing or decreasing and located above or below the x-axis. Please create a memory aid for Trisha to help her prepare for her quiz. x 2. Farah purchased a new car five years ago for $ and the car has depreciated in value by 15% per year. She would like to sell the car today in order to purchase a used vehicle for $ The used car she is intending to purchase is anticipated to retain 90% of its previous year s value each year. a. State the rule that would represent the value of Farah s original car given the number of years she has owned it. b. If Farah sells the original car, will she have the $10,000 she needs to purchase the used car? c. State the rule that could be used to model the value of the used car Farah intends to buy. d. If Farah intends to sell the used car when it is worth $6561, how long will she own it for? 172 MHS Critical Thinking and Big Ideas

23 3. Several months ago, Jonathan, Nathan and René were looking at purchasing tablet computers. Jonathan and Nathan each purchased a different tablet and René chose to invest his money because he didn t have enough to purchase either of the tablets that were available. JONATHAN S TABLET Function f represents the value of Jonathan s tablet f ( x ) = 600( 0.95) x where x: time in months since Jonathan purchased his tablet. f(x): the value of Jonathan s tablet computer in dollars. To the nearest cent, Jonathan s tablet is currently worth $419. NATHAN S TABLET Nathan s made his purchase the same day as Jonathan and his tablet was worth $450 on the day he bought it. Each month since his purchase, the tablet has decreased in value by 2% of its previous month s value. RENÉ S TABLET René did not have enough money to purchase the same tablet as Nathan so he left the $380 he had in his high interest savings account. His account pays him 1.5% interest per month. a. To the nearest cent, what is the difference in the current values of Jonathan s and Nathan s tablets? b. Does René have enough in his savings account today to purchase the tablet he wanted? MHS Critical Thinking and Big Ideas 173

24 Skill Builder 5 Solving Exponential Equations by Matching the Bases EVALUATED 1 Solve the following exponential equation by matching the bases. 7 2x = Solve the following exponential equation x ( x 2) 625 = 0 2 Solve the following exponential equation by matching the bases. 32 x = 64 (2 x) 5 Solve the following exponential equation. ( 2x+ 1) ( x+ 5) 4 27 = Solve the following exponential equation. 36 (x 3) = 216 (3 x) 182 Chapter 3: Real Functions

25 Skill Builder 6 Solve the following exponential equation. ( 7x+ 1) ( 3x+ 2) 5 25 = Estimate the value of x. 2(5 x ) = Find the zero (x intercept) of the following function. y = 4(3) (x 6) Solve the following exponential equation if possible, or, if not, estimate the value of x. 3 2x = Find the zero (x intercept) of the following function. y ( 5 x) 1 = Skill Builder: Solving Exponential Equations by Matching the Bases 183

26 CRITICAL THINKING AND BIG IDEAS Solving Exponential Equations by Matching the Bases 1. Solving exponential equations by matching the bases requires that you have a strong ability to rewrite numbers with different bases. For each of the following, rewrite them with the smallest base possible. Next, write an explanation of how to solve exponential equations by matching the bases. a. 36 = b. 243 = c. 125 = f. 343 = g = h. 121 = d. 64 = e = i. j = = Explanation: 184 MHS Critical Thinking and Big Ideas

27 2. At the beginning of math class, Paul raised his hand to say that he didn t understand the homework question from the night before. The question asked him to solve an exponential equation by matching the bases and the bases of both sides were fractions. Paul assumed that since the fraction on the left side of the equation was less than one and the fraction on the right side of the equation was greater than one there was no way of matching the bases. Paul s homework problem x 5 x = 9 8 Was Paul correct in his assumption? If not, please write a detailed explanation to Paul of how to approach problems like this and solve his problem for him. MHS Critical Thinking and Big Ideas 185

28 3. Laura is working on the exponential growth of different strains of bacteria in her science class. She is studying two different strains that are growing in separate petri dishes. STRAND 1: The number of bacteria in Strand 1 are represented by the function f ( x) x: The time in minutes since the experiment began. f(x): The number of Strand 1 bacteria present. STRAND 2: The number of bacteria in Strand 2 are represented by the function ( ) x: The time in minutes since the experiment began. h(x): The number of Strand 2 bacteria present x + = where h x = 125 x where How long into the experiment was the number of bacteria present in each strand equal? 186 MHS Critical Thinking and Big Ideas

29 Skill Builder 6 An Introduction to Piecewise Functions EVALUATED 1 Evaluate the following function at x = 3. 4 Find where f(x) = 1 in the following function 2 Find the y intercept of the following function. 5 State the domain and range of the following function. 3 Evaluate the following function at x = Chapter 3: Real Functions

30 Skill Builder 6 State the domain and range of the following function. 8 Is the following function continuous? 7 State the increasing and decreasing intervals of the following function. 9 What change would have to be made to this piecewise defined function graph to make it continuous? Skill Builder: An Introduction to Piecewise Functions 193

31 Skill Builder 10 The rule of the following function is x 1 if x < 2 f ( x) = A if 2 x 4 x 3 if x > 4 The function piece labeled A is missing Give a possible function rule for A that would make this function continuous. 194 Chapter 3: Real Functions

32 CRITICAL THINKING AND BIG IDEAS An Introduction to Piecewise Functions 1. Marcus went to the park this morning which is located 3 kilometres from his house. He left his house at 8 AM and walked to the park, which took him 15 minutes. He stayed at the park for 20 minutes and then walked home. He walked a little more slowly on the way home and it took him 20 minutes to arrive at his house. Please sketch this situation on the distance vs. time graph below. Explain what difficulties you would be confronted with in trying to determine the rule of the function representing Marcus trip to the park. MHS Critical Thinking and Big Ideas 195

33 2. Your friend Zachary has been ill for the couple days and missed the first class on piecewise functions. He has been going over the notes that you lent him, but he does not understand how to evaluate a piecewise function at a point where the domain of two pieces of the functions meet. Please write a detailed explanation and include examples for Zachary to explain how to evaluate a piecewise function at a point on a graph or by evaluating the function rule. 196 MHS Critical Thinking and Big Ideas

34 3. Veronique is planning a field trip to take her two grade 3 classes to Granby Zoo. She has just learned from their website that tickets for groups that number less than 25 people cost $30 per ticket. If there are 25 or more people in a group, the price per ticket drops to $25 per person. a. Sketch the graph of this situation. b. From analyzing your graph, determine the piecewise function rule that represents this situation Rule: MHS Critical Thinking and Big Ideas 197

35 Skill Builder 7 Graphing Piecewise Functions EVALUATED 1 Graph the following function. 1 if x < 3 f( x) = 3 if x 3 3 Graph the following function and evaluate at x = 2. f( x) = + < 4( x 2) + 4 if x > 2 4( x 2 2) 4 if x Sketch the graph of the following piecewise function. 5 if x 0 f( x) = 5 x + 5 if 0 < x Graph the following function and state its domain. 2x+ 1 if x< 1 f( x) = 2x 1 if x Chapter 3: Real Functions

36 Skill Builder 5 Graph and state the range of x+ x f( x) = x+ 4 if x> 1 2 ( 2) +2 if 1 7 Graph the following function and state its positive and negative intervals. 2 x < x if 1 2 f( x) = 2x + 8 if x > 2 6 Graph the following function and state its domain. x+ 1 if 3 x< 1 f( x) = 2 x if x> 1 8 Graph the following function and state whether it s continuous. x + x< f( x) = 5 if x 2 2 5( 2) 5 if 2 Skill Builder: Graphing Piecewise Functions 199

37 Skill Builder 9 Graph the following function and state what change would need to be made for it to be continuous. 3x 1 if x< 1 f( x) = 2 if x > 1 10 Graph the following function and state where it is undefined. x 1 if x< 2 f x = x x< x 3 if x > 4 2 ( ) ( 3) if Chapter 3: Real Functions

38 CRITICAL THINKING AND BIG IDEAS Graphing Piecewise Functions 1. Graph each of the following functions on the coordinate plane provided. 2 ( ) = 0.5x f ( x ) = 8 f ( x) 10 f x = x Explain how to take the graphs you ve created and use them to produce the graph of the piecewise function represented by: ( ) f x x, 0 x< 4 = 8, 4 x< 6 10 x, 6 x 10 MHS Critical Thinking and Big Ideas 201

39 2. Create a real-life situation that could be represented by a piecewise function. a. Represent it in words. b. State the function rule. c. Graph the piecewise function that illustrates your situation. 202 MHS Critical Thinking and Big Ideas

40 3. Graph the following piecewise function. ( ) f x 8 2 x, x< 2 = 4, 2 x< 6 0.5x + 7, x 6 a. Is the function continuous? b. Explain how to determine continuity without needing to sketch the graph. MHS Critical Thinking and Big Ideas 203

41 Skill Builder 8 Reading Step Graphs EVALUATED 1 Calculate the value of this function when x = Evaluate this function at x = 11, x = 12 and x = Evaluate f(0) in the following function Draw the open and closed dots onto the following graph given that y = 2.5 on 0.5 < x Chapter 3: Real Functions

42 Skill Builder 5 For what values of the domain does y = 6 7 Sketch the graph of the following situation for the first 2 hours 10 5 Pricing Guide First ½ hour or part of: $4 Every additional ½ hour or part of: $ Sketch the graph of the following situation for the first 3 hours. Pricing Guide First hour or part of an hour: $5 Every additional hour or part of an hour: $3 8 Sketch the function represented by the following table of values x y ]0,5] 1 ]5,10] 7 ]10,15] 13 ]15,20] 19 ]20,25] 25 Skill Builder: Reading Step Graphs 209

43 Skill Builder 9 Easy Step Parking is represented by steps below while Straight Park is represented by a straight line. Describe what factors would influence your decision to park at either of these. Cost to park your car time in hours 10 Represent the following table of values using inequality notation instead of interval notation x y ]0,50] 0 ]50,100] 2 ]100,150] 4 ]150,200] 6 ]200,250] Chapter 3: Real Functions

44 CRITICAL THINKING AND BIG IDEAS Reading Step Graphs 1. Explain how to read and evaluate a step function graph. Please provide an example and include how to deal with open and closed dots while reading the graph, as well as how to evaluate the function for a given x-value or a given y-value. 2. Chicoutimi Junior High School is offering a Spanish class to its students next year. The Principal has started looking at possible resources for the teacher and students to use in the classroom. She has decided to go with a workbook titled Everyday Spanish. The prices per student for the workbooks change according to the volume ordered. If the school orders up to 25 books, the price is $26 per workbook. If they order more than 25, up to 75 books, the price will be $22 per workbook. Finally, for orders of over 75 books, the price is $20 per book. a. Create a table of values to represent this situation. b. Sketch a step graph to illustrate the situation. MHS Critical Thinking and Big Ideas 211

45 3. Stacie is a project manager for a supply chain management company. She is responsible for organizing contracts, which includes determining budgeting and travel itineraries for all of the company s employees who need to travel to and from contracts. The company currently has a large contract at the hospital and has many employees needing to work there for at least part of each day. Stacie is evaluating the best options for them to use for parking and is looking at three possibilities. Please help Stacie write a detailed report outlining which option the employees should use depending on how long they will be at the hospital for. Include a graph in your report to allow Stacie to easily analyze the situation. Option 1: Option 2: Option 3: The parking lot at the hospital charges $5 for every hour or part of an hour. The private parking lot across the street charges a $20 entrance fee plus $2 per hour. Taking a taxi to and from the hospital costs $14 each way. Detailed Report 212 MHS Critical Thinking and Big Ideas

46 9 Associating Functions with Graphs and Tables of Values EVALUATED 1 For the following graph, find the rule of the function associated with it. Skill Builder 3 For the following table of values, find the rule of the function associated with it. x y For the following table of values, find the rule of the function associated with it. x y For the following table of values, find the rule of the function associated with it. x y Skill Builder: Associating Functions with Graphs and Tables of Values 213

47 Skill Builder 5 For the following graph, find the rule of the function associated with it. 7 For the following table of values, find the rule of the function associated with it. x y For the following table of values, find the rule of the function associated with it. x y For the following table of values, find the rule of the function associated with it. x y In this section, if the products of the ordered pairs (x, y) are constant, what type of function is this? 10 In this section, if a function increases or decreases at a constant rate, what type of function is this? 214 Chapter 3: Real Functions

48 CRITICAL THINKING AND BIG IDEAS Associating Functions with Graphs and Tables of Values 1. Martyne is studying for her upcoming quiz on Real Functions and is struggling with how to identify each of the functions she has learned about by analyzing their graphs. Complete the memory aid below to help Martyne prepare for her quiz. Linear Function Real Functions A linear function is always a straight line. If it passes through the origin, it is a direct variation and a partial variation if it does not pass through the origin. Basic Quadratic Function Basic Rational Function Basic Exponential Function MHS Critical Thinking and Big Ideas 215

49 2. Identify the characteristics of the tables of values of each of the functions from this lesson that allow you to determine which type of function is being represented. Linear Function How to Identify Functions from a Table of Values Basic Rational Function Basic Quadratic Function Basic Exponential Function 3. Provide an example of a table of values for each type of function and illustrate how to find the rule from the table of values. Function Type Table Finding the Rule Linear Function Basic Quadratic Function Basic Rational Function Basic Exponential Function 216 MHS Critical Thinking and Big Ideas

50 10 Periodic Functions Introduction EVALUATED Skill Builder 1 Draw a box around one cycle of the periodic function illustrated below. 5 Does the following illustration represent a periodic function? Justify your response. y x 2 State the frequency of the periodic function illustrated below. 6 Irena invents a new compound that when bounced will return to the same height indefinitely. Does this represent a periodic function? The pedals of a bicycle, moving at a constant speed, make 100 rotations in 25 seconds. Determine the period and frequency of a rotation. 7 Given the periodic function illustrated below, determine the value of f(41) seconds 4 The head of an oil well moves up and down 20 times per minute. State the frequency and period in both minutes and seconds Skill Builder: Periodic Functions Introduction 217

51 Skill Builder 8 Given the periodic function illustrated below, determine the value of f( 9) minutes 10 A Ferris wheel has a radius of 10 m, the center is 12 m above the ground and it rotates once every 60 seconds. Consider the periodic function f where x represents time in seconds and y represents the distance from the ground. Sketch one cycle of this function if you get on at the bottom. 9 The hour hand of a clock is 10 cm long and the center of the clock is 40 cm from the floor. Consider the periodic function f where x represents time in hours since midnight and y represents the distance between the tip of the hour hand and the floor. Sketch one cycle of this function. 218 Chapter 3: Real Functions

52 CRITICAL THINKING AND BIG IDEAS Periodic Functions Introduction 1. Pierre is having trouble understanding periodic functions and his older sister points out to him that building a periodic function is just like using the copy and paste function on a computer. a. Provide an example and describe in detail what Pierre s sister meant by this. b. Define the terms: period, cycle, and frequency. MHS Critical Thinking and Big Ideas 219

53 2. The Ferris wheel at the La Ronde amusement park in Montreal is illustrated in the image to the right. The highest point is 51 metres off the ground, the wheel has a radius of 25 metres and does one full rotation every 40 seconds. Sketch the periodic function graph of the Ferris wheel s rotation where the x-axis represents time in seconds and the y-axis represents the height of a passenger above the ground in metres (note that a passenger always gets on at the lowest possible point). On your graph, indicate the cycle, the period and the frequency of the situation. 220 MHS Critical Thinking and Big Ideas

54 3. Your friend Christiane is unable to answer one of her homework questions on periodic functions and texts you to see if you can help her. Please write a detailed explanation to Christiane to help her approach and solve her homework problem. I don t understand a periodic functions question, can you help? Sure, what s the question that s troubling you? It is asking me to find f(57), which is not on the graph that is shown. What is the domain of the graph shown and what is the period? The graph is drawn for x = [0, 20] and the period is 11 units. I ll write you a detailed explanation! MHS Critical Thinking and Big Ideas 221