In this activity, students get more practice adjusting their graphing windows. Then they explore why someone might want to choose a different graphing window to tell a different story. As students determine how graphing windows give unique meanings to the same model, students are modeling with mathematics (MP4).

The purpose of this discussion is to see how the same model can tell a different story depending on the graphing window. After demonstrating or asking a student to share how to adjust the window to display each version of the graph, focus discussion on the last three questions. Emphasize that all of the graphs are showing the same information, but that the choice of graphing window makes the information look either harmless or alarming. Here are some questions for discussion:

• “What are some reasons graphing window C might make people worry?” (It appears very steep, like the concentration of the harmful chemical is shooting up.)
• “What are some reasons someone might decide not to worry about graphing window C?” (The time span is 80 years, so perhaps the trend can be interrupted before then. Also, we would need more information about how concentrated the chemical needs to be before it is harmful. Maybe 8 parts per billion isn’t anything to worry about.)
• “What are some reasons graphing window B might look like nothing to worry about?” (It appears very flat, like the concentration of the chemical is hardly increasing at all.)
• “What are some reasons someone might decide to dig deeper if they were only presented with window B?” (It shows only 8 years, so someone might wonder what happens after that. Also, if they knew the model was exponential, they would suspect it would shoot up eventually.)

The goal of this activity is to use graphing windows to tell different stories about the same equation. Invite selected students to share their responses. Here are some questions for discussion:

• “What adjustments to the graphing window make the cost seem lower? What adjustments make the cost seem higher?” (The graph that looks quite flat and close to the horizontal axis because the graphing window has a wider range of -values than -values makes the cost seem lower. The graph that looks quite steep because the graphing window has a wider range of -values than -values makes the cost seem higher.)
• “For each situation, how did you decide whether to make the graph look flatter or steeper?” (I thought about whether the situation suggested the function should be increasing slowly or quickly.)
• “What are some strategies to make the graph look flatter?” (Keep the domain the same or smaller, and make the range larger.)
• “What are some strategies to make the graph look steeper?” (Keep the range the same, and make the domain larger.)
• “In most graphing technology, we can zoom in or zoom out. Why might we want to set a specific graphing window, instead of zooming in or out?” (Zooming in or out often doesn’t change the overall shape of the graph, because the domain and range are being scaled by the same amount. Choosing a specific graphing window gives us more control over what is shown.)