# Lesson 12

Reasoning about Exponential Graphs (Part 1)

## 12.1: Spending Gift Money (5 minutes)

### Warm-up

The goal of this warm-up is to practice modeling a situation characterized by exponential decay with an equation.

One strategy students may use to choose the right equation is by evaluating each one for some value of $$t$$. For instance, they may choose 1 for the value of $$t$$, evaluate the expression, and see if the result makes sense for the amount of money after 1 week. Though this is a valid approach, encourage students to think about what is happening in the situation and see if it is correctly reflected in the way each expression is written.

For instance, in Option A, the term $$\frac13 t$$ in the expression would mean “a third of the time in weeks” (rather than a third of the gift money), suggesting that this expression doesn’t accurately reflect the situation. Similarly, in Option C, $$\frac13 \boldcdot 180^t$$ would mean $$\frac13$$ being multiplied by 180 repeatedly ($$t$$ times), which also does not represent the situation.

### Student Facing

Jada received a gift of $180. In the first week, she spent a third of the gift money. She continues spending a third of what is left each week thereafter. Which equation best represents the amount of gift money $$g$$, in dollars, she has after $$t$$ weeks? Be prepared to explain your reasoning. 1. $$g = 180 - \frac13 t$$ 2. $$g = 180 \boldcdot \left(\frac13\right)^t$$ 3. $$g = \frac13 \boldcdot 180^t$$ 4. $$g = 180 \boldcdot \left(\frac23\right)^t$$ ### Student Response For access, consult one of our IM Certified Partners. ### Anticipated Misconceptions Some students may choose Option B because of the words $$\frac{1}{3}$$ of what is left in the prompt. Ask them what fraction of the quantity is left if $$\frac13$$ of it is spent. If students are still not convinced, ask them how much of the 180 is spent in week 1? How much is left? Use your equation to verify that 120 is left. ### Activity Synthesis Ask students for the correct response, and prompt them to explain why the other ones are not correct. Make sure students understand that if a third of the balance is spent every week, two-thirds of the balance is what remains every week, so the value of $$b$$ is $$\frac{2}{3}$$, and the amount available after $$t$$ weeks would be $$180 \boldcdot (\frac23)^t$$. ## 12.2: Equations and Their Graphs (15 minutes) ### Activity The goal of this lesson is to examine how the numbers $$a$$ and $$b$$ influence the graph representing a function $$f$$ defined by an equation of the form $$f(x) = a \boldcdot b^x$$. A money context is used in the first question to facilitate access. A bank account with more money is generally desirable so students will be looking for the function which grows most quickly. The growth of the accounts is not realistic (unless more money is being added on a regular basis), and has been chosen to motivate a close analysis of the growth factors. As students work and discuss in groups, notice those who can articulate how $$a$$ and $$b$$ in an exponential function affect a graph that represents it. Ask them to share later. This activity works best when each student has access to devices that can run the Desmos applet because students will benefit from seeing the relationship in a dynamic way. ### Launch Arrange students in groups of 2 and provide access to graphing technology. Tell students to read the first question and ask them: "If the account were yours, which function would you choose: $$f$$, $$g$$$$,$$ $$h$$, or $$j$$ ?" Give students a moment to discuss their choice and rationale with their partner. Then, tell students to read the second question and discuss with their partner which functions—$$p$$, $$q$$, or $$r$$—they would choose. Ask one partner to graph the functions in the first question and the other partner to graph the functions in the second question and then analyze their graphs together. Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. During the launch, take time to review terms that students will need to access for this activity. Invite students to suggest language or diagrams to include on a display that will support their understanding of exponential expressions in the form $$y = a\boldcdot b^x$$. Supports accessibility for: Conceptual processing; Language ### Student Facing 1. Each of the following functions $$f$$, $$g$$$$,$$ $$h$$, and $$j$$ represents the amount of money in a bank account, in dollars, as a function of time $$x$$, in years. They are each written in form $$m(x) = a \boldcdot b^x$$. $$\displaystyle f(x) = 50 \boldcdot 2^x$$ $$\displaystyle g(x) = 50 \boldcdot 3^x$$ $$\displaystyle h(x) = 50 \boldcdot \left(\frac32 \right)^x$$ $$\displaystyle j(x) = 50 \boldcdot (0.5)^x$$ 1. Use graphing technology to graph each function on the same coordinate plane. 2. Explain how changing the value of $$b$$ changes the graph. 2. Here are equations defining functions $$p$$, $$q$$, and $$r$$. They are also written in the form $$m(x) = a \boldcdot b^x$$. $$\displaystyle p(x) = 10 \boldcdot 4^x$$ $$\displaystyle q(x) = 40 \boldcdot 4^x$$ $$\displaystyle r(x) = 100 \boldcdot 4^x$$ 1. Use graphing technology to graph each function and check your prediction. 2. Explain how changing the value of $$a$$ changes the graph. ### Student Response For access, consult one of our IM Certified Partners. ### Student Facing #### Are you ready for more? As before, consider bank accounts whose balances are given by the following functions: $$\displaystyle f(x)=10\boldcdot 3^x \qquad\qquad g(x)=3^{x+2}\qquad\qquad h(x)=\tfrac{1}{2}\boldcdot 3^{x+3}$$ Which function would you choose? Does your choice depend on $$x$$? ### Student Response For access, consult one of our IM Certified Partners. ### Anticipated Misconceptions Once students have used graphing technology to graph four functions on the same set of axes, they may need help choosing a graphing window so that they can see salient features of all four graphs. They may also need help understanding how to determine which graph represents which function. (Some tools make it easier to distinguish than others.) ### Activity Synthesis Invite previously selected students or groups to share their responses. Discuss: • “On the graph, where do you see the $$a$$ of each function? How does the size of $$a$$ affect the graph?” (The $$a$$ is the $$y$$-intercept. The greater the $$a$$, the higher it is on the vertical axis.) • “On the graph, where do you see the $$b$$ of each function? How does the size of $$b$$ affect the graph?” (The $$b$$ appears in the steepness of the curve. When the function is characterized by exponential growth, the larger the $$b$$, the steeper the curve, or the more quickly the quantity is growing.) • “How is the graph of $$j$$ different than other graphs?” (It represents a situation characterized by exponential decay: $$j(x)$$ decreases as $$x$$ increases.) • “How is the $$b$$ value for function $$j$$ different than those of the other functions?” (The value of $$b$$ is less than 1.) Conversing: MLR2 Collect and Display. As students discuss their ideas with a partner, listen for and collect the language students use to identify and describe how changing the values of $$a$$ and $$b$$ affect the graph. Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions. Design Principle(s): Maximize meta-awareness ## 12.3: Graphs Representing Exponential Decay (15 minutes) ### Activity This activity complements the previous one. For a situation characterized by exponential growth, a larger base value $$b$$ means faster growth. For a situation characterized by exponential decay, however, a smaller value of $$b$$ (between 0 and 1) corresponds to faster decay. Understanding how the parameter $$b$$ influences the graph of an exponential function will prepare students to make sense of and model situations involving repeated percentage change later in the unit. ### Launch Present the following scenario: “Suppose you are presented with four functions $$m$$, $$n$$, $$p$$, and $$q$$ that describe the amount of money, in dollars, in a bank account as a function of time $$x$$, in years. If the account is yours (and more money is better), which function would you choose? Why?” Here are equations defining the functions. $$\displaystyle m(x) = 200 \boldcdot \left(\frac14 \right)^x$$ $$\displaystyle n(x) = 200 \boldcdot \left(\frac12 \right)^x$$ $$\displaystyle p(x) = 200 \boldcdot \left(\frac34 \right)^x$$ $$\displaystyle q(x) = 200 \boldcdot \left(\frac78 \right)^x$$ Give students a minute of quiet think time and ask students to share their responses. Ask a student who chose the first option and one who chose the last option to share their reasoning. Tell students they will now consider the options graphically before confirming their choice. Arrange students in groups of 2. Provide access to graphing technology. Conversing: MLR8 Discussion Supports. In groups of 2, students should take turns explaining their reasoning of how they matched the equations to the graphs. Display the following sentence frames for all to see: “The equation _____ matches graph _____ because….”, and “I noticed _____ , so I matched….” Encourage students to challenge each other when they disagree. When monitoring the discussions, point out key words and phrases such as “the bases are less than 1,” “slowly decaying,” or “more vertical curve.” This will help students solidify their understanding of how the “b” affects the graph. Design Principle(s): Support sense-making; Maximize meta-awareness Representation: Internalize Comprehension. Differentiate the degree of difficulty by beginning with more accessible values. When presenting the scenario, begin by displaying only $$m(x)$$ and $$p(x)$$ for students to consider briefly before revealing the remaining functions. In addition, some students may benefit from revisiting findings from the warm-up, in which after spending $$\frac{1}{3}$$ of the gift money, Jada had $$\frac{2}{3}$$ left. Supports accessibility for: Conceptual processing; Visual-spatial processing ### Student Facing $$\displaystyle m(x) = 200 \boldcdot \left(\frac14 \right)^x$$ $$\displaystyle n(x) = 200 \boldcdot \left(\frac12 \right)^x$$ $$\displaystyle p(x) = 200 \boldcdot \left(\frac34 \right)^x$$ $$\displaystyle q(x) = 200 \boldcdot \left(\frac78 \right)^x$$ 1. Match each equation with a graph. Be prepared to explain your reasoning. 2. Functions $$f$$ and $$g$$ are defined by these two equations: $$f(x) = 1,\!000 \boldcdot \left( \frac{1}{10} \right)^x$$ and $$g(x) = 1,\!000 \boldcdot \left( \frac{9}{10} \right)^x$$. 1. Which function is decaying more quickly? Explain your reasoning. 2. Use graphing technology to verify your response. ### Student Response For access, consult one of our IM Certified Partners. ### Activity Synthesis Focus the discussion on the connection between the numbers in the equation (especially the $$b$$) and the features of the graph. Discuss questions such as: • “In the first question, what does the point that the graphs share on the $$y$$-axis say about the situation?” (All of the accounts started with$200.)
• “How does Graph A compare to Graph D?” (Graph A appears more horizontal, so the function is decaying at a slower rate.)
• “What does the largest factor ($$\frac78$$) tell us? To which graph does it correspond?” (The function loses only $$\frac18$$ of its value when $$x$$ increases by 1, so it is decaying the most slowly. Graph A must be the graph of $$q(x) = 200 \boldcdot \left(\frac78 \right)^x$$ since it shows the slowest decay.)
• “What might a graph representing a function $$v$$ given by $$v(x) = 200 \boldcdot \left(\frac{99}{100}\right)^x$$ look like?” (It would look very close to a horizontal line because nearly all of its value remains as $$x$$ increases by 1, so it is decaying very slowly.)
• “What might a graph representing a function $$w$$ given by $$w(x) = 200 \boldcdot \left(\frac{1}{100}\right)^x$$ look like?” (It will go toward 0 extremely quickly because it loses $$\frac {99}{100}$$ of its value each time $$x$$ increases by 1, so it is decaying very quickly.)

Emphasize that, in situations characterized by exponential growth (when $$b>1$$), a larger value of $$b$$ means a curve that is more vertical. In situations characterized by exponential decay, where $$b$$ is between 0 and 1, the closer $$b$$ is to 1, the more the graph approaches a horizontal line. Conversely, the smaller the value of $$b$$, the more swiftly it heads toward 0 (the more vertical the curve is) before it flattens out and approaches a horizontal line.

## Lesson Synthesis

### Lesson Synthesis

In this lesson, we looked at how the parts of a function representing exponential change are expressed on the graph.

Display for all to see:

$$\displaystyle f(x)=a \boldcdot b^x$$

and display a blank set of coordinate axes focused on the first quadrant.

• Ask three students to propose different positive numbers.
• If $$b$$ is 2, how would these three different numbers in place of $$a$$ affect a graph representing this function? Sketch graphs. Attend to how you would decide on a graphing window.
• If $$b$$ is $$\frac12$$, what would the three graphs look like now? Sketch them.

If time permits, set a fixed value for $$a$$, ask students to suggest three different positive numbers to use in place of $$b$$, and repeat.

## 12.4: Cool-down - A Possible Equation (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

An exponential function can give us information about a graph that represents it.

For example, suppose the function $$q$$ represents a bacteria population $$t$$ hours after it is first measured and $$q(t) = 5,\!000 \boldcdot (1.5)^t$$. The number 5,000 is the bacteria population measured, when $$t$$ is 0. The number 1.5 indicates that the bacteria population increases by a factor of 1.5 each hour.

A graph can help us see how the starting population (5,000) and growth factor (1.5) influence the population. Suppose functions $$p$$ and $$r$$ represent two other bacteria populations and are given by $$p(t) = 5,\!000 \boldcdot 2^t$$ and  $$r(t) =5,\!000 \boldcdot (1.2)^t$$. Here are the graphs of $$p$$, $$q$$, and $$r$$.

All three graphs start at $$5,\!000$$ but the graph of $$r$$ grows more slowly than the graph of $$q$$ while the graph of $$p$$ grows more quickly. This makes sense because a population that doubles every hour is growing more quickly than one that increases by a factor of 1.5 each hour, and both grow more quickly than a population that increases by a factor of 1.2 each hour.