In this activity, students interpret the intersection of two graphs representing exponential equations in context. Then, given the output coordinate of the intersection, they explain why the input coordinate could be found by solving either equation separately, or by setting the equations equal to each other.

In the last question, if students are unsure of how to solve an equation with an on each side, consider asking:

• “Can you explain how you set up your equation.”

• “How could you use the properties of exponents to solve your equation?”

Focus the discussion on students’ explanations for the second question. Make sure students recall that all points on a graph representing an equation are the input-output pairs that make the equation true. Because the intersection of the two graphs, , is a point on both graphs, it is a solution to both equations. At that point, the value of and are both 171, so we can solve for by solving any of these equations:

• , or
• , or
• , or

Note that the last equation works to find the point of intersection because the graphs of the equations meet in only one point. If they met at other points, this equation would have multiple solutions.

In this activity, students use what they have learned about exponential functions to solve problems in a real-world situation. They make a prediction about whether two populations that grow exponentially will be equal at some point. They also verify a claim that one country will reach a certain population by a certain time.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5). To use an equation to verify a certain claim about a situation, students need to reason both abstractly and concretely (MP2).

Monitor for students who use these strategies to find when the functions are equal and when one of the functions reaches the target output value.

• Input different values for to find when the outputs are close.
• Solve an equation, possibly by rewriting into a logarithmic form.
• Graph the functions to find the intersection.

If students produce a graph of the functions for the first problem, observe that they do not meet for the domain they chose, and conclude that the graphs never meet, consider asking:

• “Can you explain your graph to me?”

• “How could adjusting the graphing window help you check if the two graphs meet?”

The goal of this discussion is for students to see different strategies for finding when functions are equal or when a function has a certain output value.

Display 2–3 strategies from previously selected students. Use Compare and Connect to help students compare, contrast, and connect the different strategies. Here are some questions for discussion:

• “What do the strategies have in common? How are they different?”
• “How does each strategy account for the possible negative value for in the first question?”
• “Are there any benefits or drawbacks to one representation compared to another?”

If no students graphed the two functions to answer either question, display the graphs and discuss what they allow us to see.