Lesson 1: Graphs of Piecewise Linear Functions

Transcription

1 Lesson 1: Graphs of Piecewise Linear Functions Learning Target: I can create and interpret a piecewise linear function as it models the relationship between elevation and time. Why is this relevant? There are many instances where one would need to model a two variable relationship with a graph. People who invest in the stock market interpret stock graphs in order to make decisions about which stock to invest in. Scientists create graphs from data collected during an experiment for a visual representation. Elevation vs Time Elevation is the measure of an object s distance from the ground. This can be measured in inches, feet, yards, miles, etc. Time is measured in seconds, minutes, hours, days, etc. A rising object should be represented with a positive sloped line. The faster it rises, the steeper the line. A falling object should be represented with a negative sloped line. The faster it falls, the steeper the line. A horizontal line is used to represent an object whose elevation is unchanged. 1

2 Elevation (feet) Example 1 Modeling An Elevation vs. Time Story. Our story begins when time is zero. a. A boy holds a helium balloon for 3 seconds. Is the balloon rising, falling or is elevation unchanged? b. What type of line would be used to model what is happening in part a? Draw this line on the graph below. c. At 3 seconds, the boy releases the helium balloon. Is the balloon rising, falling or is elevation unchanged? d. What type of line would be used to model what is happening in part c? Draw that line on the graph below. Time (seconds) e. When a helium balloon is released, the balloon will fly straight up in the air as long as wind does not affect its path. Explain why the line drawn to represent this does not go straight up. In other words, why isn t the line vertical? f. Why is it that we can have a horizontal line but not a vertical line in an elevation vs time graph? 2

3 Example 2. Modeling An Elevation vs. Time Graph From Watching a Video. The link below will show us a video of a man changing his elevation over a time period. After watching the video, answer the questions below: a. How high do you think he was at the top of the stairs? How did you estimate that elevation? b. Were there intervals of time when his elevation wasn t changing? Was he still moving? c. Did his elevation ever increase? When? Using the video, graph the piecewise function to model the man s elevation vs. time after answering questions d g. d. How should we label the vertical axis? What unit of measurement should we choose (feet or meters)? e. How should we label the horizontal axis? What unit of measurement should we choose? f. Should we measure the man s elevation to his feet or to his head on the graph? g. The man starts at the top of the stairs. Where would that be located on the graph? 3

4 Example 3: The elevation vs time graph below depicts a 10 minute journey of a young boy named Aaron over a ten minute time period a. What could Aaron have been doing from 0-3 minutes? b. What could Aaron have been doing from 3 minutes to 4 minutes? c. From 4 to 4.5 minutes, the line segment has a positive slope but is steeper than the line segment from 0-3 minutes. Explain what Aaron could be doing and how is it different from what he is doing in the beginning of the story? d. From 4.5 to 6 minutes, we see a horizontal line but it is a bit longer than the first horizontal line. What is happening in the story that would account for this difference in length? e. What did Aaron do from 6 to 7 minutes in this story? f. What happened from the 7 th minute to the 10 th minute? 4

5 Lesson 1 Vocabulary A single linear function is a function whose solution set makes a graph that is a straight line that goes infinitely in both directions. When the solution set of the function y = x is graphed, it looks like this: The graph of a piecewise linear function consists of two or more lines pieced together on the same graph. These line pieces might have different slopes and may or may not be connected but they will never overlap with each other. Below are some examples of piecewise linear functions. A formal definition of a piecewise linear function. PIECEWISE-DEFINED LINEAR FUNCTION: Given non-overlapping intervals on the real number line, a (real) piecewise linear function is a function from the union of the intervals on the real number line that is defined by (possibly different) linear functions on each interval What are some similarities and differences of the two graphs shown above? Discuss with a partner. An interval is a range on the x-axis from where the line begins to where it ends. The piecewise linear function on the left is made up of 3 line pieces. The first line piece has an interval of 0-2 because it begins where x = 0 and ends where x = 2. Express the other two intervals below. 5

6 The second line piece begins where x = 1. Where does it end? State the interval of this line piece. Practice Questions 1. Answer the following questions about the graph shown below. a. How many line pieces make up the piecewise linear function on the right? b. State three intervals that you see. c. State the interval of a line piece where elevation is not changing. d. State the interval of a line piece where elevation is decreasing. e. Shortly after 5 seconds, the line goes below the x-axis making elevation negative. In what real life situation could elevation be negative? f. Notice that elevation increases twice. From interval 0-2 and again from interval 8-9. During which of these two intervals is elevation increasing at a faster rate? Explain how you know this. 6

7 2. Thomas is a window washer. Starting on the ground, he climbs 6 feet up the ladder in 3 seconds. He stays there for 5 seconds to wash his first window. He climbs another 5 feet in 2 seconds. He stays there for 6 seconds to wash his second window. Then he climbs down to the ground in 3 seconds. Use the story about the window washer and model his elevation changes using the graph below. The x-axis should represent time in seconds and the y-axis should represent height in feet. Your graph should include all appropriate labels. 7

8 Lesson 2: Average Rate of Change Learning Target: I can compute the average rate of change from the graph of a line or curve on a two variable coordinate system. Why is this relevant? When a stairway is constructed in a house or building, there are strict building code regulations as to the slope of the staircase. Architects, carpenters and building inspectors must know how to compute a rate of change for a staircase so that the slope of the staircase meets regulation steepness. A staircase that is too steep would make it unsafe for someone to walk up or down to get from one floor to another. A Observe the three graphs below. B C Imagine that these lines represent staircases. Evaluate each line and determine if it would make a good staircase. Explain your reasoning. 8

9 Average Rate of Change of a Line C A The steepness of each line can be measured and assigned a value. The value is called the average rate of change. We can see that line B is the steepest line. So it should have the greatest value for an average rate of change. Line C should have the lowest value of the three. Measuring and computing the average rate of change. B rise y Average rate of change = or means change run x Together, lets find the average rate of change for line A Express the average rate of change for lines B and C. 9

10 Compare the two lines graphed on the right. Which line looks steeper? What is the average rate of change for line n? m n How about line m? Example 1. Express the average rate of change for each line graphed below: a b c d e 10

11 Finding the Average Rate of Change of a Curve. Suppose that we want to find the average rate of change from where x = 0 to x = 2. of the curve shown on the right. Example 2. For each, find the average rate of change for the interval given. [ 0, 2 ] *This is another way to state x = 0 to x = 2 [-5, 0 ] The Average rate of Change Formula There are times when you can not simply count the rise or the run because the points are too far apart on a graph. (6,190) (4,150) To express the average rate of change, you need to apply the slope formula. y x 2 2 y x 1 1 Here we have the coordinates (4,150) and (6,190). Let s label these values. x y1 x y (4, 150) (6, 190) We substitute And get

12 Example 3. The Jamison family kept a log of the distance they traveled during a trip, as represented by the graph below. a. What is the average rate of change from the 1 st hour to the 2 nd hour? b. What is the average rate of change from the 2 nd hour to the 4 th hour? c. What is the average rate of change from the 4 th hour to the 6 th hour? d. What is the average rate of change over the entire trip? e. During which interval is the average rate of change the greatest? (1) The first hour to the second hour (2) The second hour to the third hour (3) The sixth hour to the eight hour (4) The eight hour to the tenth hour f. Did you answer part e by using the slope formula or by looking at the graph? 12

13 Computing Rate of Change from a Chart x y We will use the same formula to compute and express the average rate of change between any two intervals. y Average Rate of Change = x 2 2 y x 1 1 Find the average rate of change over the interval [2,3] The interval [2,3] contains points (2,5) and (3,8). x 1 is 2, y 1 is 5, x 2 is 3 and y 2 is 8. Now substitute = Average rate of change over interval [2,3] = 3 Example 4. Use the chart to find the average rate of change over each given interval. a. [3,4] b. [4,5] c. [6,8] d. [2,5] e. [4,7] f. [2,8] 13

14 Example 5. Interpreting the Average Rate of Change In 2011, Geoffrey Mutai set the record for shortest time to complete the New York City Marathon. He finished the 26 mile race in 2 hours, 5 minutes and 6 seconds. During this marathon, Geoffrey ran the first 13 miles in 3723 seconds and 25 miles in 7214 seconds. a. Compute Geoffrey s average rate of change using the two points given above. b. If you were to graph this data, what would the average rate of change (slope) of the line connecting these two points represent in the context of this problem? Example 6. During a rainstorm, there was 1.5 inches of rain after 5 hours and 2 inches of rain after 7 hours. If you were to graph this data, what would the average rate of change (slope) of the line connecting these two points represent in the context of this problem? Lesson Summary The average rate of change is a ratio of the rise over the run. This can be found by counting rise and run between two points on a graph or by using the slope formula when two points are known. The slope formula y x 2 2 y x 1 1 can be used to find an average rate of change. 14

15 1. Practice Questions Find the average rate of change of each line or curve shown over the given interval. a. [0,6] b. [0,-3] c. [0,1] d. [0,2] e. [5,10] (10,60) (5,40) 2. What is the average rate of change of a line that starts at (3,20) and ends at (4,32)? 3. While walking up the stairs in a building, it took Steve 3 minutes to climb to the 8 th floor and 5 minutes to climb to the 14 th floor. Steve graphed an elevation vs time story of his climb up the stairs. If he connects a line between the two points given, what would the slope represent in the context of this story? 15

16 4. Calculate the rate of change for each line shown on the graph below. Line A Line B Line C Line D Line E 16

17 5. Calculate the rate of change for each line shown on the graph below. B A C D Line A Line B Line C Line D 17

18 Performance Task Answer the questions below pertaining to the graph shown. Robert is a house painter. After completing a job where he painted the exterior wall of a house, Robert noticed a few spots that he missed. So he dipped his brush in paint and began to walk up his ladder to touch up the missed spots. The graph above shows Robert s elevation in feet vs. time in seconds as he touched up a few spots. Answer each question below: 1. The graph is increasing between points A and B and between points C and D. Describe what is happening between these points. 2. What is Robert doing when the graph lines are horizontal? 3. What is Robert doing between points E and F? 18

19 4. How high up the side of the house did Robert climb when he made his first stop? 5. How long did it take for Robert to make his first climb before he stopped climbing? 6. How long did Robert stop to touch up his paint job the first time? 7. At what rate of change did Robert climb the ladder from points A to B? 8. How much higher did Robert climb up the side of the house from points C to D? 9. How long did it take for Robert to climb up the ladder from points C to D? 10. What was the rate of change between points C and D? 11. How far did Robert have to go to climb down the ladder from points E and F? 12. How long did it take for Robert to climb down the ladder? 13. At what rate of change did Robert climb down the ladder? 19

20 14. Express all key intervals for this story using time in seconds. 15. Did Robert climb up the ladder faster from points A to B or from points C to D? Why do you think that happened? 16. Suppose that when Robert was at point E, he noticed another spot 3 feet higher that needed to be touched up with paint. It takes him 5 seconds to reach that spot. Draw a line starting at point E that shows Robert s path to a new point F. Then draw a line starting at point F that would show what Robert did next. 20

21 Lesson 3: Interpreting Two Variable Graphs Learning Target: I can use a graph to answer questions pertaining to the relationship between the two variables. Why is this relevant? Graphs serve as a visual representation of a real life situation. This visual can help you to see something that you might not have been able to see from raw data. If you take the time to chart and graph personal expenses such as groceries, utilities, gasoline costs to name a few, you can save money by analyzing trends and making subtle changes grocery expense 0 January February March April May June July Rachel charted and graphed her grocery expense and noticed a spike in spending from April to May. She realized that in May she started shopping at a new store because it was a little closer to home but never realized how much more she was spending until she created and analyzed this graph. As a result, Rachel went back to her old grocery store and saved money. In Lesson 1, we interpreted graphs that related elevation to time. Graphs can also be created to relate time to other measurable quantities such as distance, speed, temperature, rainfall and more. The fundamental concepts that we have learned in Lesson 1 will apply to the graphs that we look at in Lesson 3. When a line is increasing, it means that the measurable quantity is going up over that time interval. When the line is decreasing, the quantity is going down. When the line is horizontal, the amount is unchanged or constant. 21

22 Example 1 The graph below represents a jogger's speed during her 20-minute jog around her neighborhood. a. b. Which statement best describes what the jogger was doing during the 9-12 minute interval of her jog? 1) She was standing still. 2) She was increasing her speed. 3) She was decreasing her speed. 4) She was jogging at a constant rate. How many intervals do you see where the jogger is increasing her speed? Name one. c. At which interval is the jogger increasing her speed the greatest? How do you know? d. Write a story about this jogger s experience over the course of the entire graph. 22

23 Example 2 The following graph shows the temperature (in degrees Fahrenheit) of La Honda, CA in the months of August and September of Answer the questions following the graph. a. The graph seems to alternate between peaks and valleys. Explain why. b. When do you think it should be the warmest during each day? Circle the peak of each day to determine if the graph matches your guess. c. When do you think it should be the coldest during each day? Draw a dot at the lowest point of each day to determine if the graph matches your guess. d. Does the graph do anything unexpected such as not following a pattern? What do you notice? Can you explain why it is happening? 23

24 Example 3 The following graph shows the amount of precipitation (rain, snow, or hail) that accumulated over a period of time in La Honda, CA. a. How much did it rain in La Honda, CA from August 24 th through August 30 th? b. How much rain did La Honda, CA get on August 31 st and September 1 st combined? c. On August 31 st and September 1 st, did it rain continuously or did it stop for a while and start again? How do you know this? d. How does this graph of rainfall in La Honda correspond to the graph on the previous page of temperature in La Honda? e. The term accumulate, in the context of the graph, means to add up the amounts of precipitation over time. The graph starts on August 24. Why didn t the graph start at 0 in. instead of starting at 0.13 in.? 24

25 Example 4 The following graph shows the solar radiation over a period of time in La Honda, CA. Solar radiation is the amount of the sun s rays that reach the earth s surface. a. What happens in La Honda when the graph is flat? b. What do you think is happening when the peaks are very low? c. Looking at all three graphs from examples 2,3 and 4, what do you conclude happened on August 31, 2012 in La Honda, CA? 25

26 Example 5 A graph of average resting heart rates is shown below. The average resting heart rate for adults is 72 beats per minute, but doctors consider resting rates from beats per minute within normal range. Which statement about average resting heart rate is not supported by the graph? 1) A 10-year-old has the same average resting heart rate as a 20-year-old. 2) A 20-year-old has the same average resting heart rate as a 30-year-old. 3) A 40-year-old may have the same average resting heart rate for 10 years. 4) The average resting heart rate for teenagers steadily decreases. Lesson Summary. A story graph relates time to a measurable quantity like distance, speed or temperature. An increasing line over a time interval. A decreasing line over a time interval. A horizontal line over a time interval. indicates an increase in the measurable quantity indicates a decrease in the measurable quantity indicates no change in the measurable quantity 26

27 Speed (MPH) Practice Questions 1. The graph below models a car s speed vs. time. Answer the questions that follow. b a c Time (seconds) a. Which line piece (a, b or c) represents a time when the car s speed was decreasing? How do you know this? b. Explain what was happening over the time interval represented by line piece b. c. At what time(s) was the car not moving? d. How many seconds did it take for the car to go from 0 to 60? 27

28 Lesson 4: Graphing Two Stories Learning Target: I can create and interpret a graph of a story that involves two variables. Why is this relevant? Recall Rachel in the previous lesson. Rachel used the visual representation of a graph to determine that changing her supermarket was hurting her budget. Rachel would not be able to use a graph to make decisions if she did not know how to create and interpret such a graph. Using Rate of Change to Create and Graph a Story Example 1. Jack Harrison had to walk to work one day because his car was in the shop being repaired. He started from home and walked to work at a consistent rate of 3 miles per hour. Follow the directions below and answer each question. a. Label the x and y axis. b. c. Indicate on the graph with an x where Jack would start. Explain why this is a good starting point. d. Identify Jack s rate of change. e. f. g. h. i. How can we use the ratio to model Jack s distance over time? Create the graph that displays Jack s distance over time. According to your graph, how far has Jack walked after 2 hours? Jack works 10 miles from home. Approximately how long did he walk? What will your line look like when Jack gets to work? 28

29 Graph Scaling Often times, you need to make decisions about how you are going to number your x and y axis. This decision will be based on the data that you have to work with and the number of gridlines that exist on your graph. For example, you may choose to skip by 10 s on your x-axis because the data contains high values and you want to represent these values. Below are some examples of graphs with varying scales. A B C D You may choose your scaling but you must stay consistent on the same axis. This means that you can not skip by 2 at first and then later skip by 5. Notice in graph D that the x-axis and y-axis have different scales, which is fine. Discuss why this might be necessary. The broken line: The y-value data numbers on the graph on the right contain the values 23,24, 27,31 and 33. The creator of this graph wanted to skip by ones but did not wish to begin at 0 since the lowest number was 23. So a broken line was used to indicate inconsistency. Example 2. Given the x and y data values, scale the graph on the right appropriately. Plot the points. X values: 3, 8, 11, 16, 18, 21, 22 Y values: 6, 14, 18, 25, 29, 38, 42, 51, 56 29

30 Example 3. Raymond just finished working at the movie theater and was using his bicycle to go home. He rides his bike at a constant rate of 500 feet per minute. The movie theater is 3000 feet from his house. Answer the questions below and graph this story on the grid provided. Label the y-axis distance from home. a. How would you label the x-axis? b. What scale would you use for the x-axis? c. What scale would you use for the y-axis? d. When time is zero, what is Raymond s distance from home? e. What is the average rate of change for this story and how do we express that on our graph? f. According to your graph, how many minutes does it take Raymond to get home? 30

31 Example 4. Graphing two stories at once. Consider the story: Duke starts at the base of a ramp and walks up it at a constant rate. His elevation increases by 3 ft. every second. Just as Duke starts walking up the ramp, Shirley starts at the top of the same 25 ft. high ramp and begins walking down the ramp at a constant rate. Her elevation decreases 2 ft. every second. a. Label and scale the x and y axis. b. c. Sketch two graphs on the same set of elevation-versus-time axes to represent Duke s and Shirley s motions. What are the coordinates of the point of intersection of the two graphs? At what time do Duke and Shirley pass each other? 31

32 Example 4. Two cars are travelling north along a road. Car 1 travels at a constant speed of 50 mph. Car 2 starts at the same time that car 1 starts, but car 2 starts 100 miles farther north than car 1 and travels at a constant speed of 25 mph throughout the trip. a. Label and scale your x and y axis. Graph the story for both cars. b. Discuss ways in which this situation is different from the situation in example #3. c. At what time does car 1 begin? How about car 2? d. What is the average rate of change for car 1? Car2? e. Do the cars ever cross paths? If so, what is the point of intersection? 32

33 Example 4 continued. Suppose that car 1 travels at a constant speed of 25 mph. Car 2 starts at the same time that car 1 starts, but car 2 starts 100 miles farther north than car 1 and also travels at a constant speed of 25 mph throughout the trip. f. Label and scale your x and y axis. Graph the story for both cars. g. How is this story different from the story on the previous page? h. Notice that these lines are parallel (do not intersect). What causes this to happen? 33

34 Practice Questions 1. Have you heard the story, The Tortoise and the Hare? This is a story of two race cars competing in a 500 mile race. Car A can go 160 miles per hour and car B can go 120 miles per hour. They re off!!!!! Both race cars ride at maximum speed for 1 hour. Car A breaks down and it takes the driver 1.5 hours to fix it. Car A is back on the track and rides at maximum speed until he crosses the finish line. Car B never stopped from start to finish. a. b. c. Label and scale your x and y axis so that you are able to display the entire race for both cars on your graph. Model the story on the graph below. During the first hour of the race, what is the average rate of change for car A? Car B? d. e. What type of a line did you draw for car A during the time when the driver was repairing the car? Approximately how long does it take car B to complete the race? f. When car B finishes the race, approximately how many miles has car A driven? g. Who wins the race? 34

35 Elevation Lesson 5: Non-Linear functions Learning Target: I can interpret and create non linear graphs which model natural situations in our environment. Why is this relevant? Possessing the ability to create a visual model of a natural occurrence can help you to be successful in other classes that you currently take or will take. In addition, a seismologist might need to create a graph which relates energy waves over a short period of time in order to learn about effects of certain earthquakes. Example 1. Watch the videos linked below a. What is happening to the speed of these objects as they get closer to the ground? b. Describe how the elevation is changing throughout the story of an object falling towards the ground. c. Draw an elevation vs time graph of an object falling towards the ground Time 35

36 Man walking down the stairs Elevation vs Time. Linear Object dropped from the roof of a building Quadratic Example 2. Answer the questions below about an elevation-versus-time graph of a ball rolling down a ramp. a. From the time of about 1.7 sec. onward, the graph is a flat horizontal line. If Ken puts his foot on the ball at time 2 sec. to stop the ball from rolling, how will this graph of elevation versus time change? b. Estimate the number of inches that the elevation of the ball changed from 0 sec. to 0.5 sec. Also estimate the change in elevation of the ball between 1.0 sec. and 1.5 sec. 36

37 The Exponential Model Example 3. The chart below is a doubling chart. Notice how each y value is double the previous y value. a. Plot the points from the chart onto the graph. Determine an appropriate scale for both the x and y axis. x y Living cells can double over time. When bacteria grows, it starts with one cell and doubles quickly in reproduction. In a matter of minutes, one cell can become millions of cells. This type of relation is called an exponential relation. Example 4. Doubling is a key word that implies an exponential relation. Other words that imply an exponential relation is tripling, quadrupling or halving. Illustrate each of these relations using the charts below: a. Tripling b. Quadrupling c. Halving x y x y x y

38 Example 5. a. Create a chart of the number of bacteria versus time for the following story: Dave is doing an experiment with a type of bacteria that he assumes divides in half exactly every 30 min thus causing the amount of the bacteria to double. He begins at 8:00 a.m. with 5 bacteria in a Petri dish and waits for 2 hrs. At 10:00 a.m., he decides the amount of bacteria is too large of a sample and adds a chemical to the dish, which takes a half an hour to kill 90% of the bacteria. One hour later, the bacteria was triple of what it was at 10 a.m.. Time (x) # of bacteria (y) 8:00 8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 1:00 1:30 2:00 b. If Dave had not added that chemical at 10:00 a.m., what would have been the bacteria count at 2:00p.m.? c. Was the chemical effective in reducing the amount of bacteria long term? 38

39 Population Practice Questions 1. Below are three stories about the population of a city over a period of time and four population-versus-time graphs. Two of the stories each correspond to a graph. Match the two graphs and the two stories. Write stories for the other two graphs. Story 1: The population size grows at a constant rate for some time, then doesn t change for a while, and then grows at a constant rate once again. Story 2: The population size grows somewhat fast at first, and then the rate of growth slows. Story 3: The population size declines to zero. Draw a graph of the story that did not match a graph above. Time 39

40 2. Create a story to match each graph below: 40

41 3. For each story graph shown below, determine if it represents a piece-wise linear function, a quadratic function or an exponential function. A B C time time time a. Which of the three graphs above could represent a ball falling to the ground? b. Which of the three graphs above could represent a person driving to work, turning around to come almost all the way back home and finally driving back to work at a faster rate? 4. Explain why a person walking down three stories of a flight of stairs could be modeled by a linear representation while an object dropped from the third story window to the ground would be modeled by a quadratic representation. 41

42 1. Standardize Test Preparation Lessons 1-5 Firing a piece of pottery in a kiln takes place at different temperatures for different amounts of time. The graph below shows the temperature in a kiln while firing a piece of pottery after the kiln is preheated to 200 o F. During which time interval did the temperature in the kiln show the greatest average rate of change? (1) 0 to 1 hour (3) 2.5 hours to 5 hours (2) 1 hour to 1.5 hours (4) 5 hours to 8 hours 2 During a recent snowstorm, Jaime noted that there were 4 inches of snow on the ground at 3:00 p.m., and there were 6 inches of snow on the ground at 7:00 p.m. If she were to graph this data, what does the slope of the line connecting these two points represent in the context of the problem? 42

43 3. To keep track of his profits, the owner of a carnival booth decided to model his ticket sales on a graph. He found that his profits only declined when he sold between 10 and 40 tickets. Which graph could represent his profits? 4. A driver leaves home for a business trip and drives at a constant speed of 60 mph for 2 hours. Her car gets a flat tire, and she spends 30 minutes changing the tire. She resumes driving and drives at 30 miles per hour for the remaining one hour until she reaches her destination. On the graph below, draw a graph that models the driver s distance from home. 43

44 5. The graph below shows the variation in the average temperature of Earth s surface from During which 5-year interval did the temperature variation change the most? Explain how you determined your answer. 44

45 6 The graph below models Craig s trip to visit his friend in another state. In the course of his travels, he encountered both highway and city driving. Based on the graph, during which interval did Craig most likely drive in the city? Explain your reasoning. 45