Lesson 16

This activity prepares students to interpret and compare functions in upcoming activities. Students are given only broad descriptions about two functions and no specific information, and then asked to interpret equations and inequalities written in function notation.

Invite students to share their interpretations of the given statements. Make sure students understand that the 3, 6, and \(m\) represent the number of months, and that at some number of months \(m\), the balances in the accounts are equal. Discuss questions such as:

• “Which account do you think started out with more money?”
• “Which account do you think has more money after 6 months?”
• “What might be reasonable values for \(m\)? How do you know?” (Between 3 and 6. At 3 months, investment account \(f\) had more money, but after 6 months it has less, so at some point between 3 and 6 months the balance of investment account \(g\) exceeds that of account \(f\).)

Then, select some students to share their graphs and check that the graphs meet the criteria of the three given statements. Highlight that the graphs of the functions could be very different, depending on the assumptions we make about how the accounts are changing.

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.

Display the graph for all to see and ask students to read the opening statement. To help students relate the equations to the graphs before they begin, ask questions like:

• “Which population covers a larger area when the experiment begins?” (Population A. Its starting value is 24 square millimeters. The initial value of Population B is 9 square millimeters.)
• “Which bacteria population area is growing more quickly?” (Population B because \(e^{(0.6h)}\) grows faster than \(e^{(0.4h)}\)since it has a larger exponent.)
• “What is the approximate growth factor per hour of each population?” (About 1.49 for Population A because \(e^{0.4} \approx 1.49\) and about 1.82 for Population B because \(e^{0.6} \approx 1.82\).)

Students may be unsure how to start the last question where they have to solve an equation with an \(e\) on each side. Remind these students that \(e\) is just a number and that the properties of exponents can help them re-write the equation so there is only a single \(e\) value.

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, \((h, 171)\), is a point on both graphs, it is a solution to both equations. At that point, the value of \(f(h)\) and \(g(h)\) are both 171, so we can solve for \(h\) by solving any of these equations:

• \(f(h) = 171\) or \(24 \boldcdot e^{(0.4h)}=171\)
• \(g(h) = 171\) or \(9 \boldcdot e^{(0.6h)}=171\)
• \(f(h) = g(h)\) or \(24 \boldcdot e^{(0.4h)} = 9 \boldcdot e^{(0.6h)}\)

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 with minimal scaffolding. 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).

As students work, look for those whose strategies are similar to those shown in the Student Response. Highlight these approaches during discussion.

Students may 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. Ask these students how they know that the two functions do not meet for some other, larger input value. 

Make sure students see that each problem could be approached in more than one way. If no students graphed the two functions to answer the first question, display the graphs and discuss what they allow us to see.

The claim in the second question could be verified by evaluating the function at the given input, or by solving the equation that represents the function when its value is 30 million. Be sure students see both pathways.