Lesson 13

Reasoning about Exponential Graphs (Part 2)

13.1: Which One Doesn’t Belong: Four Functions (5 minutes)

Warm-up

This warm-up encourages students to look closely at functions and articulate the ways in which they are similar or distinct. There are many possible responses to the question. Given their current work on exponential functions, students might be inclined to wonder if the equations define exponential functions, to look for a growth factor, or to think about the initial value of the function.

Student Facing

Which one doesn’t belong?

\(f(n) = 8 \boldcdot 2^n\)

\(g(n) = 2 \boldcdot 8^n\)

\(h(n) = 8 + 2n\)

\(j(n) = 8 \boldcdot \left( \frac12 \right)^n\)

Student Response

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Activity Synthesis

Invite students to share their responses. Highlight ideas that are important to this unit, for example, how the vertical intercept for each function can be identified, that \(j\) is a function involving exponential decay, etc. Encourage precision in students' mathematical language, for instance, by reinforcing the use of terms such as exponential, growth factor, initial amount, base, exponent, linear, etc.

13.2: Value of A Computer (15 minutes)

Activity

In a previous lesson, students used equations defining functions to reason about graphs. Here they work in the opposite direction, analyzing graphs and creating corresponding equations. In the first question, the coordinates given are not of two consecutive values of \(x\), so students need to reason about the decay factor indirectly. They may:

  • Try different factors that, when multiplied by \(f(0)\) twice, give \(f(2)\).
  • Use the graph to estimate \(f(1)\), and then use that estimate to find a reasonable decay factor.
  • Find the factor that takes \(f(0)\) to \(f(2)\) (or \(f(2)\) to \(f(4)\)), recognize that number as \(b^2\), and then find \(b\).

The second question has no context and is thus more abstract. This prepares students to interpret graphs in the next activity, where minimal information is given.

Launch

Arrange students in groups of 2. Ask students to take a few minutes of quiet think time on each question before discussing their thinking with a partner. 

Reading, Conversing, Writing: MLR5 Co-Craft Questions. To help students make sense of an exponential situation in which the graph is given before the equation, start by displaying only the first task statement and its graph. Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to share their questions with the class, then reveal the questions. Listen for and amplify any questions seeking to write an equation, interpret the meaning of various points, or predict future trends. This will help students create questions and use language that demonstrates what they know is true about a situation given a graph first.
Design Principle(s): Maximize meta-awareness; Cultivate conversation

Student Facing

  1. Here is a graph representing an exponential function \(f\). The function \(f\) gives the value of a computer, in dollars, as a function of time, \(x\), measured in years since the time of purchase.
    Graph of a decreasing exponential function, x y plane, origin O.

    Based on the graph, what can you say about the following?

    1. The purchase price of the computer
    2. The value of \(f\) when \(x\) is 1
    3. The meaning of \(f(1)\)
    4. How the value of the computer is changing each year
    5. An equation that defines \(f\)
    6. Whether the value of \(f\) will reach 0 after 10 years
  2. Here are graphs of two exponential functions. For each, write an equation that defines the function and find the value of the function when \(x\) is 5.

    1. Graph of an increasing exponential function, x y plane, origin O. 
 
    2. Graph of a decreasing exponential function, x y plane, origin O.

Student Response

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Student Facing

Are you ready for more?

Consider a function \(f\) defined by  \(f(x)=a\boldcdot b^x\).

  • If the graph of \(f\) goes through the points \((2,10)\) and \((8,30)\), would you expect \(f(5)\) to be less than, equal to, or greater than 20?
  • If the graph of \(f\) goes through the points \((2,30)\) and \((8,10)\), would you expect \(f(5)\) to be less than, equal to, or greater than 20?

Student Response

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Anticipated Misconceptions

If students have trouble getting started, ask them to consider the graph and say what they know is true about the situation that the graph represents. Most likely, they will answer some parts of the first question by doing this, and will be in a better position to figure out any questions they didn't answer.

Activity Synthesis

Select students to share their responses to the first question. Focus the discussion on how students found the decay factor for the computer. Make sure students understand that the key is to notice that every 2 years, the computer's value is multiplied by \(\frac{1}{4}\). This means that the annual decay factor is \(\frac{1}{2}\), since \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\).

For the last graph, discuss questions such as:

  • “What information is needed to describe an exponential function?” (the factor of growth or decay and the initial value)
  • “Where on the graph do we see the initial value?” (the \(y\)-intercept of the graph)
  • “What points on the graph can we use to help us find the growth factor?” (the points \((\text-1, 40)\) and \((0,20)\), or\((0,20)\) and \((2,5)\))
  • “Why might \((\text-1,40)\) and \((0,20)\) be a more strategic choice than \((0,20)\) and \((2,5)\) for finding the factor of growth?” (The \(x\) coordinates of first two points differ by 1, so the quotient of the \(y\) coordinates of those points can be used to find the factor of decay. It is a more direct way to find the factor of decay, which is the factor by which the output of \(f\) changes when the input \(x\) changes by 1. In the second pair, the \(x\) coordinate changes by 2.)
Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, display the equation \(100 = 400\boldcdot (b)^2\) and use color or annotations to illustrate how each value or variable is represented graphically.
Supports accessibility for: Visual-spatial processing; Conceptual processing

13.3: Moldy Wall (15 minutes)

Activity

This activity presents students with two graphs without numerical values assigned to any points on the graphs. The only labeled point is where the two graphs intersect. Students will need to use their understanding that tripling each day is a greater growth factor than doubling each day. This is their only means to decide which function represents which situation. Students then interpret the graphs in terms of the situations that they represent.

Launch

Display the graph for all to see. Give students a moment to observe the graph and ask, "What do you notice? What do you wonder?" Invite students to share their observations and questions.

Student Facing

Here are graphs representing two functions, and descriptions of two functions.

Graph of two exponential functions, origin O. time(months) and area (square inches). 
  • Function \(f\): The area of a wall that is covered by Mold A, in square inches, doubling every month.
  • Function \(g\): The area of a wall that is covered by Mold B, in square inches, tripling every month.
  1. Which graph represents each function? Label the graphs accordingly and explain your reasoning.
  2. When the mold was first spotted and measured, was there more of Mold A or Mold B? Explain how you know.
  3. What does the point \((p, q)\) tell us in this situation?

Student Response

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Anticipated Misconceptions

Some students may think that the dashed graph represents the mold that is tripling because it has a greater vertical intercept. Ask these students to think carefully about the second question.

Activity Synthesis

Select students to share their responses and explanations. Focus on how they deciphered the meanings of the graphs. Discuss questions such as:

  • The solid graph started off lower than the dashed graph. What does that mean in this context? (The mold population represented by the solid graph is initially smaller or covers a smaller area on the wall than the one represented by the dashed graph.)
  • After some point, the solid graph passes the dashed one. What does that mean in this context? (The mold population that started out covering a greater area on the wall is surpassed by the other population, so it is growing at a slower rate.)
  • What does the intersection of the two graphs mean? What do \(p\) and \(q\) represent? (The two mold populations are the same at time \(p\) with population covering an area of \(q\) square inches.)
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each explanation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

In this lesson, we interpreted graphs and reasoned about their equations. Show the graph of \(y =f(x)\) for an exponential function \(f\) with some points labeled. 

Graph of exponential decay function. Points P, Q, and R are labeled on graph. Origin labeled O. Horizontal axis labeled x. Vertical axis labeled y.

Tell students that the  \(x\)-coordinate of \(Q\) is 1 and the \(x\)-coordinate of \(R\) is 2. Ask students what information would give us insight into the situation and help us write an equation corresponding to the graph.

  • “What do you need to know in order to describe how the function is decreasing?” (coordinates of two of the labeled points)
  • “If you could find out the coordinates of two of these points, which two would you choose? Why?” (\(P\) and \(Q\) or \(Q\) and \(R\) because the quotient of their \(y\)-coordinates gives the growth factor.)
  • “How would you find the equation using these two coordinates?” (use the \(y\)-coordinate of \(P\) and the growth factor)

Consider showing the graph of \(y=f(x)\) along with the graph of \(y=g(x)\) for another exponential function \(g\).

Graphs of 2 exponential decay functions. Blue graph labeled f and green graph labeled g. Origin labeled O. Horizontal axis labeled x. Vertical axis labeled y.

Ask students to compare the two functions:

  • “Which function is decreasing faster? How do you know?” (\(g\) because it has a larger \(y\)-intercept than \(f\) but then takes smaller values than \(f\) as \(x\) grows)
  • (If the graphs cross) “What does the intersection of the two graphs mean?” (The outputs of \(f\) and \(g\) are the same for this value of \(x\).)

13.4: Cool-down - Two Graphs (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

If we have enough information about a graph representing an exponential function \(f\), we can write a corresponding equation. Here is a graph of \(y = f(x)\).

An equation defining an exponential function has the form \(f(x) = a \boldcdot b^x\). The value of \(a\) is the starting value or \(f(0)\), so it is the \(y\)-intercept of the graph. We can see that \(f(0)\) is 500 and that the function is decreasing.

Graph of a function on grid.

The value of \(b\) is the growth factor. It is the number by which we multiply the function’s output at \(x\) to get the output at \(x+1\). To find this growth factor for \(f\), we can calculate \(\frac{f(1)}{f(0)}\), which is \(\frac{300}{500}\) or \(\frac35\). So an equation that defines \(f\) is:

\(\displaystyle f(x) = 500 \boldcdot \left(\frac{3}{5}\right)^x\)

We can also use graphs to compare functions. Here are graphs representing two different exponential functions, labeled \(g\) and \(h\). Each one represents the area of algae (in square meters) in a pond, \(x\) days after certain fish were introduced.

  • Pond A had 40 square meters of algae. Its area shrinks to \(\frac{8}{10}\) of the area on the previous day.
  • Pond B had 50 square meters of algae. Its area shrinks to \(\frac 25\) of the area on the previous day.
Graph of 2 functions on a grid.

Can you tell which which graph corresponds to which algae population?

We can see that the \(y\)-intercept of \(g\)'s graph is greater than the \(y\)-intercept of \(h\)'s graph. We can also see that \(g\) has a smaller growth factor than \(h\) because as \(x\) increases by the same amount, \(g\) is retaining a smaller fraction of its value compared to \(h\). This suggests that \(g\) corresponds to Pond B and \(h\) corresponds to Pond A.