Activity
A flag ceremony is held at a Fourth of July event. The height of the flag is a function of time.
Here are some graphs that could each be a possible representation of the function.
a.
b.
c.
d.
e.
f.
After you examine the graphs, discuss with a partner how each graph could be used to describe the movement of the flag as it is raised on the flagpole.
1. Either select one of the graphs above or use the one assigned to you by your teacher. Explain what the graph tells us about the flag and its height over time.
Compare your answer: Your answer may vary, but here are some samples.
- Graph A: The flag starts out on the ground but is immediately raised at a constant rate.
- Graph B: The flag starts out at 3 meters above ground and then stays at that height the entire time.
- Graph C: The flag starts out on the ground but gets pulled up quickly. The movement slows down and there is a brief pause, and then it is raised up again at a quicker and quicker rate.
- Graph D: The flag starts out on the ground, is raised up a couple of meters, and then is lowered back to the ground. This happens repeatedly.
- Graph E: The flag starts out at 0.5 meter above the ground and is raised up bit by bit, with a brief pause before it is raised farther.
- Graph F: The flag starts out on the ground and is raised up at a constant speed for 3 seconds. It stops abruptly and comes back down at a constant speed. Then, 2 seconds later, it stops at 3 meters above the ground.
2. Which graph seems the most realistic and likely to happen at a flag ceremony? Be prepared to explain your reasoning.
Compare your answers: Your answer may vary, but here is a sample.
Graph E seems the most realistic. It starts at a height where someone would link the flag to the rope and then it rises in spurts like the movement that might happen as someone pulls on the string.
3. Which graphs seem most unrealistic or least likely to happen at a flag ceremony? Be prepared to explain your reasoning.
Compare your answers: Your answer may vary, but here is a sample.
Graphs B and D are the least realistic. On graph B, the flag seems to be attached to the flagpole at one height and then it is never moved up the flagpole. On graph D, the person who raises the flag seems to put it on the ground and then raise it, lower it, raise it, lower it, etc.
4. Which graphs might be possible, but seem unlikely? Be prepared to explain your reasoning.
Compare your answers: Your answer may vary, but here is a sample.
Graphs A, C, and F are possible, but it is unlikely that the flag starts out being on the ground.
Use the following graph to answer questions 5 - 6. Note that the graph also relates time and height.
5. Can this graph represent the time and height of the flag? Be prepared to show your reasoning.
Compare your answer: Your answer may vary, but here is a sample.
If we assume that time is measured in seconds and the height is measured in meters, this graph would not represent the height of a flag raised at a flag ceremony. The graph shows the flag being at all heights between 0 and 6 meters at the same time (3 seconds), which is impossible.
6. Is this a graph of a function? Be prepared to show your reasoning.
Compare your answer: Your answer may vary, but here is a sample.
This graph does not represent a function. A function has only one output for every input. If we assume that time is measured in seconds and the height is measured in meters, this graph shows many possible outputs when the input is 3 seconds.
Are you ready for more?
Extending Your Thinking
Suppose an ant is moving at a rate of 1 millimeter per second and keeps going at that rate for a long time. If time, , is measured in seconds, then the distance the ant has traveled in millimeters, , is . If time, , is measured in minutes, the distance in millimeters is .
1. Explain why the equation gives the distance the ant has traveled, in millimeters, as a function of time, , in years.
Compare your answer: Your answer may vary, but here is a sample.
For example: The number of seconds in a year is . If the ant travels 1 millimeter a second, in one year, it would travel millimeters. In years, it would travel millimeters.
2. Use the Desmos graphing tool or technology outside the course to graph the equation.
3. Does the graph look like that of a function? Why do you think it looks this way?
Compare your answer: Your answer may vary, but here is a sample.
The graph is a vertical line and doesn’t look like a graph of a function. The slope is really large, so the line is so steep that it looks vertical.
4. Using graphing technology, adjust the graphing window until the graph no longer looks this way. If you manage to do so, describe the graphing window that you use.
Compare your answer: Your answer may vary, but here is a sample.
The graph is a slanted line when the -axis is scaled from –0.0001 to 0.0001 years and the -axis is scaled to ten thousands in both directions.
5. Do you think the last graph in the flag activity, Graph F, could represent a function relating time and height of a flag? Explain your reasoning.
Compare your answer: Your answer may vary, but here is a sample.
It could, depending on the unit of time being used and the scale of the horizontal and vertical axes. If the time is measured in months, for instance, each point on the graph could represent a time value that is just a tiny bit more than 3 months—say, 3 months and 1 second, 3 months and 2 seconds, and so on. A horizontal change of 1 second would not be visible when the horizontal units are measured in months, so all input-output pairs marking the height of the flag at each second would appear to be stacked exactly on top of one another.
Video: Interpreting Graphs Without Units
Watch the following video to learn more about how to determine if a graph is realistic when it doesn’t have units.
Self Check
Additional Resources
Interpreting Graphs of Functions
The graph below represents the total number of smartphones that are shipped to a retail store over the course of days.
Match each part of the graph (A, B, and C) to its verbal description. Be prepared to explain the reasoning behind your choice.
| Description | Details |
| Description 1 | Half of the factory workers went on strike, and not enough smartphones were produced for normal shipments. |
| Description 2 | The production schedule was normal, and smartphones were shipped to the retail store at a constant rate. |
| Description 3 | A defective electronic chip was found, and the factory had to shut down, so no smartphones were shipped. |
Description 1 best matches part C of the graph. If half of the workers went on strike, then the number of smartphones produced would be less than normal. The rate of change for C is less than the rate of change for A.
Description 3 best matches part B of the graph. If no smartphones are shipped to the sore, the total number remains constant during that time.
Part A of the graph best matches description 2. If the production schedule is normal, the rate of change of interval A is greater than the rate of change of interval C.
Try it
Try It: Interpreting Graphs of Functions
The relationship between Jameson’s account balance and time is modeled by the graph below.
Write a story that models the situation represented by the graph.
Compare your answer: Your answer may vary, but here is a sample:
Jameson was sick and did not work for almost a whole week. Then, he mowed several lawns over the next few days and deposited the money into his account after each job. It rained several days, so instead of working, Jameson withdrew money from his account each day to go to the movies and out to lunch with friends.