The purpose of this activity is to show a method of determining the exponential equation from a graph.

This activity uses the Worked Example routine. In this routine, students analyze steps used to solve a problem and discuss specific steps with a partner and the whole class. During the discussions, students construct viable arguments to justify the steps taken by the problem solver (MP3). Later, students can apply these strategies to solve similar problems.

The purpose of this discussion is to make connections between the parts of an exponential equation and where they are seen in the graph of the equation.

Tell students to open their books or devices. Display the following questions, and give students 1–2 minutes of quiet think time to consider their responses.

• “How did the solver know this would be an exponential equation?” (The graph does not look like a straight line, and the coordinates that are provided do not subtract the same value each time the -coordinate increases by 1, so it is not linear. The line curves in the same way as exponential graphs, so checking to see if there is a common growth factor makes sense.)
• “Why did the solver do so many division problems to figure out the growth factor? How did they know which points to use?” (If the solver knew this was an exponential equation, they would need to divide only two of the points to calculate the growth factor (although checking more would confirm that their growth factor was correct). In this problem, the solver did not know for sure and needed to check multiple pairs of points to make sure they had the same growth factor. To calculate the growth factor, the solver should use points that are 1 -value apart. This gives the number multiplied by the first point to get to the next point.)
• “How did the solver write the equation? How can they check to see if they are correct?” (Since there was a constant growth factor of 0.8, the solver confirmed that this equation is exponential, so they could write an exponential equation using the growth factor and vertical intercept. They can check by testing the points they know in the equation or testing other points they can see on the graph.)

Ask partners to compare their responses to each question and decide if either or both answers are correct. During this partner discussion, monitor to select groups who reason about the questions in different ways. Follow with a whole-class discussion, inviting 1–2 selected groups to share their answers for each question.

If time allows, consider displaying a second worked example next to the first that either:

• Shows a different method of solving a similar problem that students can compare and contrast with the first example (MP7)
• Shows the same method with an error that students can critique and correct (MP3)

This activity consists of two problems that are quite similar. After giving students a chance to tackle them on their own, it may be desirable to demonstrate one, and then ask students to complete the other independently. Students look for and make use of the structure of the equations, graphs, and tables to solve problems and make connections (MP7).

In this partner activity, students take turns finding a graph that represents a given function. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).