1. Functions 6 Identifying Key Information from the Graph Warm-up 1 For each of the following, highlight the regions of the graphs that correspond to different graph shapes (constant, increasing, or decreasing). Use three different color markers or highlighters. = Constant = Increasing = Decreasing (color) (color) (color) 2. 3. 4. 5. 6. 8. 7.
Activity 1 1. How many different ways can you draw a continuous increasing function connecting point A to point B? All graphs must represent functions. That is, there can be no strictly-vertical or left movements as you move from A to B. Mathematicians know of at least three primary types. Can you think of more? Draw as many as you can below.
Activity 1 2. Match each graph to a particular runner based on the description of each runner s behavior. Write at least two sentences to describe how you determined that the graph matched the runner s description. Each graph describes the distance the runner has traveled versus time since they started running or walking. A. Jay runs cross-country for his high school team. He decides to go for a jog around the park on Tuesday afternoon. Luckily for him, there were not many people at the park and he could run at a comfortable pace for an hour. B. Jessie usually walks to school in the mornings. Today she did not check the time when she left her house. As she started walking, she realized that she was late and sped up until she sprinted to get to her class on time. C. Jun decided to go for a run as his exercise for the day. He started out sprinting around his neighborhood, but fatigue kicked in and he had to slow down until his run became a walk, and eventually slowed to a stroll. C is the correct response. This graph shows distance changing quickly at the beginning. For example, over the first 10 minutes the distance changed from zero to more than 2 miles. During the last 10 minutes, from 50 to 60 minutes the distance changed less than one mile, from about 3.75 to 4, which indicates a much slower speed.
Activity 1 A is the correct response. This graph shows a constant rate of change. To check this, notice that between time 0 and 10 minutes the distance changed 1 mile. Between times 40 and 50 the distance went from 3 to 4 miles, again a 1 mile difference, which indicates a constant rate of change. B is the correct response. In the beginning, close to time 0 the distance does not change fast. You can see this by noticing that within the first 5 minutes the change in distance is only about a quarter of a mile. Towards the end, say the last 5 minutes the distance changes more, from about 2 to 2.5 miles, which is more, and indicates a faster pace.
Exit Pass 1 Fill in the following table. Choose from the 7 Verbal Descriptions at the bottom of the page and for your example make up a story that would match the shape of the graph. Label the x- and y- axes with appropriate units (words only, no numbers needed). Graph Verbal Description Example Increasing at an increasing rate Something like B above. Decreasing at a decreasing rate (careful, as x increases the rate becomes more and more negative so it is decreasing more and more) Increasing at a constant rate. Ball falling from a height. Profit made selling lemonade, assuming an initial cost to pay for the booth. Decreasing at an increasing rate (as x gets larger the rate becomes less and less negative, hence rate is increasing) Constant Glass of hot tea placed outside on a freezing day (in Celcius). Price to mail up to a one ounce letter. Increasing at a decreasing rate. Decreasing at a constant rate. Profit from the stock market from 2010. Note in 2008 the stocks took a dive, resulting in a loss, but stocks rose steeply from the low and then settled into less and less % growth each year. Distance from your house as you walk at a constant pace and go by your house on the way to a friend s house. Here negative distance means you re past your house. Verbal Descriptions: Decreasing at a constant rate Increasing at a decreasing rate Decreasing at a decreasing rate Increasing at a constant rate Increasing at an increasing rate Decreasing at an increasing rate Constant
Warm-up 2 Vocabulary toolbox: Constant Increasing Decreasing Linear Increasing at an increasing rate Decreasing at a decreasing rate Increasing at a decreasing rate Decreasing at an increasing rate Increasing at a constant rate Decreasing at a constant rate Describe the shapes of the different parts of the graphs below using the vocabulary toolbox. Mention specific points of interest in your descriptions. More than one behavior may apply at a time. Note: designation changes at x = -2, 0, 1.5, 3, 4.5, 6, and 8 Note: Designation changes at x = -4, 2, 8, 9, and 11.
Activity 2 Reading Qualitative Graphs Determine which function graph matches the statement. Indicate why the other graphs do not match the statement. State a possible function domain (input values) and range (output values), and any possible restrictions to either or both. 1. A school bus drives 30 miles per hour for awhile and then pulls into the school and lets off students. The third graph from the left is the correct one. The first one fails because it never has speed 0. The second fails because it did not have the constant speed for awhile. The last one fails because it ends with positive speed, the students could not be let off. Domain = from zero to 5, or when the bus stops. Range from zero to 30. Can t have negative speed nor speed greater than 30 mph. 2. A bungee jumper jumps off a 100-foot high bridge. The graph on the far right is correct. The first on the left fails because is begins with height zero and then gets bigger, and there is no oscillation. The second graph fails because it has negative height. The third graph fails because it does not oscillate. Domain must be positive, say 0 to 10 seconds, or from 0 to when he stopped. Range is from the lowest point he goes, which is not 0 to the beginning height. 3. A bicyclist climbs up a hill at a steady pace of 10 miles per hour, and speeds up as she glides down the other side. The graph on the far right is the correct one. The first two fail because the speed is not constant in the first part. The third one fails because the speed actually got less at the end. Domain = zero to when gets to the bottom. Range = 10 mph to say 17 or 18 mph according to the graph.
Activity 2 Reading Qualitative Graphs 4. An anteater at the zoo is given ants for lunch. He eats 4 ants and takes a break, repeating this pattern throughout lunch. The graphs below depict the number ants remaining, beginning when the ants are delivered. The second graph from the left is the correct one. The first fails because there are no pauses. The third is not a function. The one on the far right fails because it has an increasing population of ants. Domain = zero to 3 minutes or whenever he finishes lunch. The range according to the graph is some positive number through the number of ants he was given to eat. It would be acceptable to say the range was 0 to some positive value. Note: the range should only be positive integers, although the graph does not show this. 5. A lawn service mows the grass every three weeks. The graph on the far left is the correct one. The second one fails since it is increasing, and therefore never shows the grass getting cut. The third one fails because the grass appears to get cut slower than it then grows. The fourth one never shows the grass growing again. Note: the graph that works should show longer growing times in comparison to being cut. As it is it appears to take days to cut the grass. Domain = zero to 12 weeks, or 0 to 84 days. Range = 1 inch to 2.5 inches or thereabouts. Note: there are a lot of answers for domain and range that are acceptable. The main points are that some values are not allowable, such as negative height, and that the domain is made up of the x-coordinates and the range is made up of the y-coordinates. Also, the range for #4 consists only of integers.
Warm-up 3 Match the graph to the appropriate data summary. Once matched, add the appropriate values and units for the x- and y- axes. 1 2 3 4 A. The number of tourists at Hapuna Beach changes throughout one day. The park opens at 8am and closes at 6pm. Notable increases are seen before lunch and dinner hours. (Number of tourists vs. Time of day) Graph 3 (similar to graph 1, except for the times of day where the increases occur). X-units should be hours with each tick mark approximately one hour. Y-values are # of people in the park with each tick mark approximately 100 people, although anything between 50 and 200 would be fine. B. The average for Honolulu Airport shows increases of flights for work rush hours of 7 a.m. and 5 p.m., and greater lulls at 2 p.m. and after 8 p.m. (Number of flights vs. Time of day) Graph 1. X-units are hours, with each tick mark approx. 3 hours. Y-units are number of flights per hour with each tick mark approximately 5. C. NASCAR racers travel around a track and stop an average of 6 times per 1000-kilometer races. The race lasts about 4 hours and they stop at the same place each time on the track. (Distance from race finish line vs. Time) Graph 4. X-units are time with each tick mark 30 minutes (remember it is a 4-hour race, but the graph only shows about half the race since there are only 3 stops showing). Y-values are miles with each tick mark about 25 km (the total race is 1000 km, so they need to travel about 167 km between stops. D. Surfers paddle out to the break and catch an average of 5 waves within an hour. (Distance from shoreline vs. Time) Top right. X-units are time with each tick mark about 5 6 minutes. Y-units are distance in ft from shore with each tick mark some reasonable number so that the maximum from shore is reasonable.
Activity 3 Graphs to Stories On the following page you will find four graphs. Below are 4 scenarios that go along with the four graphs. For each graph you are to write a more detailed story that is consistent with the various parts of the graph and answer the question(s). Be sure to include appropriate units! (The labels on the axes tell you which graph goes with which story.) 1. Twins Jack and Jill need to each write 20 thank-you notes for their birthday presents. Write a story about what each of them did to get their job done, according to the graphs. Based on the graph, which twin finished their thank-you notes first and how fast did each twin write their notes? Jack started later than Jill, but he wrote each of his slower so he finished first. Jack began writing his letters 5 hours after Jill began. 10 hours after Jill started Jack caught up to her, at that point in time they had completed the same number of letters, approximately 11. 2. Marley started a pet boutique in the mall. He used his own money to start-up the business, but eventually started making a profit. Describe a possible scenario of what Marley did in order that his profits resulted in the graph. Be sure to include what he does during days 0 3. During which 5 days does Marley make the most profit? How do you know? What might he have done on those 5 days? Marley lost money the first 4 days, although less and less each day. This is because he advertised and treated his customers good and more and more came in or returned. He made a profit on day 5 but lost money again on day 6 because it was a Saturday and less customers came on Saturdays each week. He ran specials beginning on day 6 and he made more and more each day based on his good work. The second Sat his profits went down, but they were still positive. After day 12 he ran even more specials and word of mouth got around and his profits continued to rise, except for the next Saturday. 3. No Coqui! opened their business with the claim that they could exterminate a guaranteed number of coqui frogs per day from any location with a Coqui infestation. Describe a particular Coqui infestation based on the graph. Include how many coqui frogs they began with, how many frogs the company exterminated per day, and how long it took to completely eliminate the Coqui frogs. They began with 100 frogs and eliminated the same number of frogs each day, about 15 per day. It appears they will be able to eliminate the entire population around day 8. 4. The Earth Activist Club noticed that the beach was increasingly covered with litter. They organized a one-month cleanup campaign at the beach that resulted in the graph provided. Describe their plan and the results, and explain if the Earth Activist Club was successful and how you determined this. When they began, on a Saturday, there were about 80 visible pieces of trash on the beach. They eliminated about half the first day. They did not return until the following Saturday. During the week the trash got worse again, but only increased to about 65 pieces. They eliminated all but about 30, but when they returned the next week it had gone up again to 45. Each week there was less trash when they came back and it appears the campaign is successful, since the last time they came they were able to remove all but about 5 pieces.
Activity 3 Graphs to Stories
Formative Assessment Create a story based on the graph provided. Include labels for units and values on the x- and y- axes of each graph. Write enough detail in the story so that the points of interest are acknowledged and are coherent with the story. Story 1: Story 2: Story 3: