Using Graphs to Convey Information

In the first four problems, some factor is changing over time. Sketch graphs of each on some graph paper and answer the questions. In the second four problems, decide what in the problem could be illustrated with a graph and build a graph.

Problem 1: Hansel and Gretel

Hansel and Gretel leave their house and start walking towards their grandmother's house. It is exactly 1 mile to grandmother's house and the entire trip takes about 2 hours (including the 30 minutes spent eating brownies with grandmother). You know the following information about the trip:

• Once Hansel and Gretel start walking, they do not change their speed. They can stop, though.
• After 15 minutes (1/4 of an hour) of walking, Hansel and Gretel were 1/4 of the way to grandmother's house.
• After 30 minutes (1/2 of an hour) of walking, Hansel and Gretel were 1/2 of the way to grandmother's house.
• After 45 minutes (3/4 of an hour) of walking, Hansel and Gretel were 3/4 of the way to grandmother's house.
• Hansel and Gretel reached grandmother's house after exactly one hour of walking.
• When Hansel and Gretel had been at their grandmother's house for 30 minutes, they realized that they had only 30 minutes to walk back home in time for dinner, and leave grandmother's house to start walking back home.
• They arrived home exactly thirty minutes after they left grandmother's.

On a coordinate plane, construct a graph of Hansel and Gretel's journey with distance as the dependent variable and time as the independent variable. Did they walk faster on the way to grandmother's house or on the way back from grandmother's house?  How is this illustrated on your graph?

Problem 2: Roller Coaster

You and a group of your friends have gone to an amusement park for the day and decide to ride Thunder Bolt, a new roller coaster ride. You notice that the roller coaster moves significantly faster on some parts of the ride than on others. Using the following information, construct a graph of the ride with velocity as the dependent variable and time as the independent variable. Show the change in the roller coaster's speed over the first 7 seconds of the ride.

• The roller coaster begins at a constant speed of 5 m/s and is travelling uphill. This constant speed is maintained for the first four seconds of the ride.
• In the next second, the roller coaster slows down (the speed decreases at a constant rate) as you reach the top of the hill. The roller coaster reaches the top of the hill exactly five seconds into the hill and stops momentarily before it begins its descent.
• As the roller coaster speeds down the hill, its speed increases continuously for exactly 2 seconds, reaching a maximum of 10 m/s at this point.

What does your graph look like over the interval t=0 seconds to t=4 seconds? What does this mean about the roller coaster's speed over this interval? The roller coaster's speed is decreasing between t=4 and t=5 and increasing between t=5 and t=7. How is this increase and decrease illustrated on your graph (is the slope of the graph positive or negative over each of these intervals)?

Problem 3: Marathon Runner

Your friend ran in an 8 mile marathon last weekend. Using the following information, construct a graph of her run with distance as the dependent variable and time as the independent variable.

• In the first two hours, she ran at a constant speed of 2 miles per hour.
• In the third hour, however, your friend was tired and took an hour break.
• In the fourth hour, she ran twice as fast as in the first two hours to make up for the time she lost during her break.

How far did your friend run in the first two hours of the race? How far did she run in the fourth hour of the race? How does your graph illustrate that she ran twice as fast in the fourth hour as in the first two hours?

Problem 4: Race Car Driver

Your uncle is a race car driver. One day you decide to go watch a race.  Your uncle began the race well but got a flat tire 2.5 minutes into it and had to stop.  You collected the following data while you were watching the race:

 Time Distance 0 0 0.5 0.5 1 1 1.5 1.75 2 3 2.5 4.5 3 4.5 3.5 4.5 4 4.5 4.5 4.5 5 4.5

Construct a graph of your uncle's race with distance as the dependent variable and time as the independent variable. What was happening to your uncle's speed (not distance) between t=0 and t=2.5? Between t=2.5 and t=5? How does your graph illustrate these changes in speed?

Problem 5: Trip to School

Andrew rides his bike to school every day. He is usually not fully awake by the time he leaves, so it takes him time to gain his speed. After about 5 minutes he reaches the stop sign at the end of his road, and has to pause for a minute. He is going to get to school early so he just peddles along for a couple of moments. Suddenly his friend Dietrich comes up behind him and they start racing. They are both going pretty fast, but then Andrew remembers he left his science project on the table at home. Andrew quickly turns around and speeds back home to get it. Now he has only 6 minutes until the start of school, and is going to be late if he doesn't hurry. Andrew pedals as fast as he can all the way back to school and makes it to class just in time for the first bell.

Problem 6: Foxes and Rabbits

Jason is an amateur naturalist and is counting the number of foxes and rabbits he sees in the park near his house. On his first time out he finds 200 rabbits but only 30 foxes. When he goes out to count the next year he finds that the foxes have eaten 75 rabbits and the fox population has increased to 50. The next year he counts only 25 foxes and the rabbit population is down to 50. Jason is concerned that something is happening to the rabbits and foxes. When he goes out the next year he finds there are 150 rabbits but the foxes are down to just 10. In his fifth year of counting he finds 200 rabbits and 30 foxes just like in the first year. Jason starts to see that as the number of rabbits goes up the foxes eat more. That means the number of rabbits goes down and the foxes goes up. But as the rabbits go down farther the foxes start to go down as well. With fewer foxes now the rabbit population can start to go up and start the cycle again.