Lesson 16

Using Graphs and Logarithms to Solve Problems (Part 2)

16.1: Two Bank Accounts (5 minutes)

Warm-up

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.

Student Facing

A business owner opened two different types of investment accounts at the start of the year. The functions \(f\) and \(g\) represent the values of the two accounts as a function of the number of months after the accounts were opened.

  1. Here are some true statements about the investment accounts. What does each statement mean?
    1. \(f(3) > g(3)\)
    2. \(f(6) < g(6)\)
    3. \(f(m) = g(m)\)
  2. If the two functions were graphed on the same coordinate plane, what might it look like? Sketch the two functions.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

16.2: Bacteria in Different Conditions (20 minutes)

Activity

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.

Launch

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 \(e^{0.6} \approx 1.82\).)
Conversing, Writing: MLR5 Co-Craft Questions. Use this routine to spark students’ curiosity about the graphs that represent the growth of two populations of bacteria. Display only the image of the graph and ask students to write down possible mathematical questions that could be asked about the graph. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions about the intersection of the graphs. For example, “What is the point of intersection between the exponential functions?” This will help students create the language of mathematical questions themselves before feeling pressure to produce answers. 
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, remind students that this task contains all familiar skills but perhaps in a combination that students may not recognize right away. Remind them that they have practiced converting equations using \(e\) into their natural logarithm form. They also have the ability to solve systems of equations and isolate variables, which they mastered in previous units. If students are intimidated by combining these skills for the first time, check in with them to help them recognize the prior skills as they work.
Supports accessibility for: Social-emotional skills; Conceptual processing

Student Facing

To study the growth of bacteria in different conditions, a scientist measures the area, in square millimeters, occupied by two populations.

The growth of Population A, in square millimeters, can be modeled by \(f(h) = 24 \boldcdot e^{(0.4h)}\) where \(h\) is the number of hours since the experiment began. The growth of Population B can be modeled by \(g(h) = 9 \boldcdot e^{(0.6h)}\). Here are the graphs representing the two populations.

Coordinate plane, x, time, hours, 0 to 7, y, populations, square millimeters, 0 to 250. Two exponential functions. Line A begins 0 comma 24, line B begins 0 comma 9, line B meets line A near x = 5.
  1. In this situation, what does the point of intersection of the two graphs tell us?
  2. Suppose the population coordinate of the point of intersection is 171. Explain why we can find the corresponding time coordinate by:
    1. solving \(f(h) = 171\) or \(g(h)=171\)
    2. solving the equation \(f(h) = g(h)\)
  3. Solve either \(f(h) = 171\) or \(g(h)= 171\). Show your reasoning.
  4. Solve \(f(h) = g(h)\). Show your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

The functions \(f\) and \(g\) are given by \(f(t) = 10e^{0.5t}\) and \(g(t) = 8e^{0.4t}\).

  1. Is there any positive value of \(t\) so that \(f(t) =g(t)\)? Explain how you know.
  2. When do \(f\) and \(g\) reach the value 1000?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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.

16.3: Populations of Two Countries (10 minutes)

Activity

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.

Launch

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy as they compare populations and answer the questions. For example, “I predict _____ because. . .” and “How do you know. . . ?”
Supports accessibility for: Language; Social-emotional skills

Student Facing

The population, in millions, of Country C is given by the equation \(f(t) = 16 \boldcdot e^{(0.02t)}\). The population of Country D is given by \(g(t) = 17.5 \boldcdot e^{(0.025t)}\). In both equations, \(t\) is the number of years since 1980.

  1. Will there be a time when the two populations are equal? Explain or show your reasoning.
  2. At some point in time, the population of Country C reached 30 million. When does this happen? Explain or show your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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. 

Activity Synthesis

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.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion about the strategies for finding the year when the population of Country C reached 30 million. After students find the year when the population reached 30 million, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their strategies. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast the strategies for finding the value of \(t\) so that \(f(t)=30\). This will help students understand and find connections between multiple approaches to solving this problem. 
Design Principle(s): Cultivate conversation

Lesson Synthesis

Lesson Synthesis

The goal of this discussion is to emphasize that graphs and equations can both be used to compare functions and find when they reach certain values or when they both have the same value.

Display this example for all to see: Suppose we have two functions modeling the growth of two types of cells as a function of time in days. The function modeling the number of Type A cells is defined by \(A(d) = 400 \boldcdot e^{(0.03d)}\) and the function modeling the number of Type B cells is defined by \(B(d) = 300 \boldcdot e^{(0.045d)}\).

Then, discuss questions such as:

  • “How can we tell which cell type is growing faster?” (From the equations, we can see that Type B has a greater growth factor. If we graph the functions, we can see that the graph of B shows a quicker increase per day.)
  • “If we graph the two functions, they intersect. What do the coordinates of the intersection point tell us?” (It tells us the time and number of cells when there are the same number of Type A and Type B cells.)
  • “How might we find out on which day there will be the same number of cells of each type?” (We can estimate from the graph, or we can solve the equation \(A(d) = B(d)\).)

If time allows, ask students to solve the equation \(400 \boldcdot e^{(0.03d)}= 300 \boldcdot e^{(0.045d)}\) to determine what day there will be the same number of cells. (about 19.2 days)

16.4: Cool-down - Reaching 40 Thousand (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Graphs representing functions can help us visualize how two or more quantities are changing in a situation. Let’s consider the populations of two colonies of ants.

The population, in thousands, of a colony of carpenter ants and a colony of red wood ants can be modeled with functions \(c(x) = 8.1 \boldcdot e^{(0.03x)}\) and \(r(x)= 5.4 \boldcdot e^{(0.05x)}\), respectively. Here, \(x\) is the time in months after the colonies were first studied.

From the equations, we can tell which colony had a greater initial population (carpenter ants, 8.1 thousand) and which had a greater growth factor (red wood ants, \(e^{(0.05)}\)). Will the colony of red wood ants eventually exceed that of the carpenter ants? If so, when might it happen? Graphs representing \(y = c(x)\) and \(y = r(x)\) can help us answer these questions.

Coordinate plane, 2 exponential functions.

The intersection of the graphs tells us that about 20 months after the study began, the two colonies have the same population, about 15 thousand. After that point, the population of red wood ants is greater than that of carpenter ants. To find out more exactly when the two colonies have the same population, we can use graphing technology to find the coordinates of the intersection point more precisely.

Another way to find the point of intersection is using the equations for the functions. At the point of intersection of the graphs, the two functions have the same \(y\)-value, so we can write the equation \(8.1 \boldcdot e^{(0.03x)}=5.4 \boldcdot e^{(0.05x)}\). Then we can solve this equation:

\(\displaystyle \begin {align} 8.1 \cdot e^{(0.03x)} &= 5.4 \cdot e^{(0.05x)}\\ \frac{8.1}{5.4}&=\frac{e^{(0.05x)}}{e^{(0.03x)}}\\ 1.5&=e^{(0.02x)}\\ \ln (1.5)&= 0.02x\\ \frac{0.405}{0.02}& \approx x\\ 20.3& \approx x\\ \end {align}\)

This solution means that about 20.3 months after the study began, the two colonies have the same population.