19.1: Enough with the Zooming Already (10 minutes)
Provide access to graphing technology. It may be desirable for students to work in groups of four, so that they can view all four graphing windows on four different devices at the same time.
If needed, demonstrate using technology to graph the function and set the graphing window for the first given window. Then, allow students to set the remaining graphing windows and answer the questions.
Use graphing technology to create a graph of \(y=10^x\). Here are some different graphing windows to try:
- A: \(\text-10 \le x \le 10\) and \(\text-10 \le y \le 10\)
- B: \(\text-1 \le x \le 1\) and \(\text-10 \le y \le 10\)
- C: \(\text-50 \le x \le 50\) and \(\text-10 \le y \le 10\)
- D: \(\text-1 \le x \le 1\) and \(\text-50 \le y \le 50\)
- Which graphing window makes the graph look . . .
- the steepest?
- the flattest?
- Come up with a new graphing window that makes the graph look even steeper than the steepest one you identified.
- Come up with a new graphing window that makes the graph look even flatter than the flattest one you identified.
The main purpose of this activity is to learn or revisit how to specify a graphing window using the available technology. Once students show proficiency with that, quickly move on to the next activity.
19.2: How to Lie with Graphing Windows (20 minutes)
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.
Display the 4 graphs for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.
These graphs represent a function modeling the amount of a harmful chemical in drinking water in parts per billion over time in years.
Here is Graph A, with rectangles approximating the graphing windows of B, C, and D superimposed. Which rectangle matches which graph?
- Use graphing technology to create a graph of \(f(x)=0.2 \boldcdot (1.045)^x\). Practice adjusting the graphing window so your graph looks like A, B, C, and D.
- Imagine you are a public health official worried about this model, and you want to convince others that they should be worried, too. Which graphing window would you use, and why?
- Imagine you are a public relations official for the company responsible for the chemical in the drinking water, and you want to convince others not to worry. Which graphing window would you use, and why?
- What questions should a journalist writing about the issue ask each person about their graphs, and which graph should she publish with the article?
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 the choice of graphing window makes the information look either harmless or alarming. 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 actually 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 only shows 8 years, so someone might wonder what happens after that. Also, if you knew the model was exponential, you would suspect it might shoot up eventually.)
19.3: Renting a Car (10 minutes)
In this activity, students choose a graphing window to tell a particular story.
Continue access to graphing technology. Students may be ready to dive into the activity without much assistance, or may need to engage in a notice and wonder to familiarize themselves with the context and the information encoded in the equation and the graph.
Suppose \(y=0.50d + 4\) represents the cost of renting a car, \(y\), as a function of miles driven, \(d\). Here’s a graph representing the function.
- What is the graphing window used for the given graph?
- Find a graphing window so that:
- It gives the impression that the charge per mile driven is very high and the total rental cost gets really expensive really fast.
- It gives the impression that the charge for every mile driven is close to nothing and the total rental cost will be pretty low even if the car is driven many miles.
Invite selected students to share their responses. Questions for discussion:
- “What adjustments to the graphing window make the cost seem lower? What adjustments make the cost seem higher?” (The graph in the second question make the cost seem lower, because the graph looks quite flat and close to the horizontal axis. The graph is the first question make the cost seem higher, because the graph looks quite steep.)
- “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 were some strategies to make the graph look flatter?” (Keep the domain the same or smaller, and make the range larger.)
- “What were some strategies to make the graph look steeper?” (Keep the range the same, and make the domain larger.)
- “In most graphing technology, you can zoom in or zoom out. Why might you 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 allows you more control over what is shown.)