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Algebra 2

A Solidify Understanding Task

Unit 1 | Lesson 3

Tracking the Tortoise

MATERIALS

graphing calculators or graphing technology such as GeoGebra or Desmos

Printable Student Materials (PDF format)

Printable Student Materials (docx format)

REQUIRED PREPARATION

Anticipate Student Thinking

To prepare for the lesson, work the task and think about these questions:

What strategies for recognizing and finding inverse functions should students be comfortable with at this point?
How will you help students make sense of the notation for logarithms to describe the inverse of an exponential?
How will you use this lesson to create a conceptual foundation for logarithms that can be extended in the next unit to graphing logarithms and using logarithms to solve problems?
What strategies might students use that are not in the Anticipating and Monitoring chart?
What misconceptions do you think students may have that are not in the Anticipating and Monitoring chart?

PROGRESSION OF LEARNING

�This unit introduces the relationship between functions and their inverses using story contexts that are modeled by the functions that students are most familiar with from previous courses, linear, quadratic, and exponential. This lesson supports students in making sense of the inverse of an exponential function by using story context, tables, and graphs before discussing the notation for a logarithm. Students’ flexible work with representing inverses will allow them to find an inverse for an exponential function, without being able to write an equation. This is a critical lesson in the development of the concept of logarithms. It helps students to connect the relationship between exponential and logarithmic functions to the relationship between other familiar functions and their inverses, using multiple representations. It is not the purpose of this lesson to do any significant work with logarithms, only to introduce the term and general idea. As the unit progresses, students will be writing inverse functions for the three different types of functions and using representations to identify inverse functions. In Unit 2, students will do extensive work with logarithms and logarithmic functions.

�PURPOSE�

The purpose of this lesson is:

To extend the concept of inverse developed through work with representations on linear and quadratic functions to exponential functions.
To develop informal ideas about logarithms based upon understanding inverses. These ideas will be extended and formalized in Unit 2.
To build on students’ experience with representing inverses to understand a defining feature of inverse functions: If 𝑓(𝑎) = 𝑏, then 𝑓⁻¹(𝑏) = 𝑎

The first part of the task builds on earlier work in Algebra 1 with exponential functions. Students are given an exponential context, the distance traveled for a given time, and then asked to reverse their thinking to examine the time traveled for a given distance. After creating a graph of the situation, students are asked to consider the extended domain of all real numbers and to think about how that affects the inverse function. Problems at the end of the task press on understanding of the inverse relationship and the idea that a function and its inverse "undo" each other, using function notation and particular values of a function and its inverse.

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What are we learning and doing today?

LEARNING GOALS

Introduce the inverse of an exponential function as a logarithmic function.
Solidify understanding of the relationship between a function and its inverse with focus on generalizing that if then .

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EXPLORE

If , what is ? How do you know?

If , what is ? Will your answer change if is a different function? Explain.

x	y
3	8
1/2	1.414
a	b

EXPLORE (20 MINUTES)

NARRATIVE

As students are working individually, see that they have a good idea of how to get started with problems 1 and 2 after the Launch. Problem 2 asks for a description of the graph, which could be accomplished by a quick sketch. If students do not know the shape of an exponential graph, encourage them to use graphing technology to see the graph of the function and then sketch it on their papers. After they have the shape in mind, they should write a few features, such as the graph is always increasing, it has no x-intercept, and it contains the point (0, 1). Once most students have completed problem 2, ask them to share their work with their partners and then complete the task working with partners or small groups.

MONITOR STUDENT THINKING

In problems 3 and 4, students determine the time given a distance. Solving the equations directly does not require logarithms, since they can think about powers of 2 to get the solutions. If they look at an equation and don’t know how to solve it, ask what power of 2 gives 64 for an answer. They may need technology to find this, either numerically or graphically. The answers for problems 3 and 4 are designed to get students started on building a table of values to graph the function in problem 5.

A common misconception in problems 5 is that time must always be considered the independent variable. Although time is often the independent variable, there is no reason that it must be. If students are having difficulty thinking about time as the input, ask if they have ever cheered someone running a marathon. Typically, bystanders choose a location to watch the runner and then clock the time when the runner went past. In this case, students are not finding the location based on the time, they are finding the time based on the location.

In problem 6, students find the time for a distance that is not a power of 2. Solving the equation directly would require logarithms, which students have not been introduced to. The purpose of this problem is to get students to wonder about the function that undoes an exponential. They can estimate the solution using their graphs to produce an acceptable answer.

Problem 8 can be done using graphing technology. Help students to see that when considered on its entire domain, f⁡(x) = 2x has a horizontal asymptote, and so f⁻¹⁡(x) will have a vertical asymptote. Students will probably not use the word asymptote because they have not had much experience with it, but they should be aware of the behavior of the graph near the axes. This question also reinforces ideas from previous tasks about the relationship between the graph and the domain and range relationship for functions and their inverses.

Support for Students with Disabilities

Action and Expression: Physical Action

Provide access to graphing calculators or graphing technology. Some students may benefit from guidance including steps for using the graphing tools.

� Provides accessibility for:

Organization
Conceptual Processing
Attention

Problems 9–11 are all looking for the same idea, the concept that inverse functions undo each other. Problem 9 starts the idea with the context of the task, and problems 10 and 11 extend it. Students may have difficulty with reading the notation. If this is the case, support students in translating the statements one piece at a time. For instance, translate the statement f⁡(a) = b to be, "the output of the function at a is b."

SELECT AND SEQUENCE STUDENT THINKING

Begin by selecting a student for problems 1 and 2 who has calculated some values and created a sketch of the graph of d⁡(t). For problem 5, pick a student who has graphed the inverse by switching the inputs and outputs in the table of d⁡(t) to obtain t⁡(d). Identify a student to share problem 8 who has accurately graphed both functions. Choose a student for problems 9 and 10 who has an explanation for why if f⁡(3) = 8, then f⁻¹⁡(8) = 3.

NOTE

Refer to your Teacher Edition: 5 Practices Charts book for preparation and guidance around in-class discussions.

NOTE FOR EVALUATING RESPONSES

If 𝑓(3) = 8, what is 𝑓⁻¹(8)? How do you know?

𝑓⁻¹(8) = 3

If 𝑓(½) = 1.414, what is 𝑓⁻¹(1.414)? How do you know?

𝑓⁻¹(1.414) = ½

If 𝑓(𝑎) = 𝑏, what is 𝑓⁻¹(𝑏)? Will your answer change if 𝑓(𝑥) is a different function? Explain.

𝑓⁻¹(𝑏) = 𝑎

This is true for all functions based on the definition of an inverse.

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Takeaways

5 Minutes

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Exponential Function and Its Inverse—table

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Inverse function form graph

This slide is hidden from presentation view and is intended as a Note for Evaluating Responses. You may choose to unhide the slide and present to students by right clicking on the slide and choosing “unskip slide”.

NOTE FOR EVALUATING RESPONSES�

Exponential functions and their inverses:

The inverse of an exponential function is a logarithmic function.
Both exponential and logarithmic functions can have different bases. The base of the logarithm will match the base of the exponential. In our case, f⁡(x) = 2^x and f⁻¹⁡(x) = log2⁡^x.
The graph of a logarithmic function is a reflection over the y=x line of the exponential function with the same base.

Inverse functions undo each other:

if f⁡(a) = b, then f⁻¹⁡(b) = a.

Example: f⁡(x) = x² and f⁻¹⁡(x) = √x

f⁡(2) = 2²= 4 and f⁻¹(4) = √4 = 2

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Inverse from Equation

This slide is hidden from presentation view and is intended as a Note for Evaluating Responses. You may choose to unhide the slide and present to students by right clicking on the slide and choosing “unskip slide”.

NOTE FOR EVALUATING RESPONSES�

Exponential functions and their inverses:

The inverse of an exponential function is a logarithmic function.
Both exponential and logarithmic functions can have different bases. The base of the logarithm will match the base of the exponential. In our case, f⁡(x) = 2^x and f⁻¹⁡(x) = log2⁡^x.
The graph of a logarithmic function is a reflection over the y=x line of the exponential function with the same base.

Inverse functions undo each other:

if f⁡(a) = b, then f⁻¹⁡(b) = a.

Example: f⁡(x) = x² and f⁻¹⁡(x) = √x

f⁡(2) = 2²= 4 and f⁻¹(4) = √4 = 2

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GLOSSARY

logarithmic function �(logarithm)

The inverse of an exponential function is called a logarithmic function.

If , then .

The base of the log and the base of the exponent match. A logarithm has 3 parts: the argument, the base, and the answer.

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TAKEAWAYS

DISCUSS (30 MINUTES)

ROUTINE

Pause and Record

NARRATIVE

The focus of the discussion will be exploring the inverse of an exponential function. Students are ready to discuss when they have completed problem 9.

CONNECT STUDENT THINKING�

Begin the discussion by asking a student to present a table of values for the graph of t⁡(d). Ask students to describe how they selected the values of d to elicit the idea that they chose powers of 2 because they made it easy to think about the time. This strategy will become increasingly valuable in the next unit, so it is important to make it explicit here. Ask a student to share the graph of the inverse function t⁡(d), and ask the class how it compares to the graph of d⁡(t). They should again notice the reflection over y = x. Ask students to compare the domain and range of d⁡(t) with the domain and range of t⁡(d). As students discuss the graphs and conclude that t⁡(d) is the inverse function of d⁡(t), ask them to begin to generalize the patterns they see about inverses to all functions. Some questions to ask are:�

Will a function and its inverse always be reflections over the y = x line? Why?
Will the domain of the function always be the range of the inverse function? Why or why not?

Move the discussion to problem 8. It is important in this early exposure to the ideas of logarithms that students consider the part of a logarithmic graph between x = 0 and x = 1. As students draw f⁻¹⁡(x), have them consider some particular values in this region. Use the strategy of finding powers of 2 to generate values for x = ½, ¼, ⅛. Use their answers from the previous part of the discussion to extend the graph and justify the vertical asymptote at x = 0. Tell students that this is a new kind of function that will be explored in depth later, but it is called a logarithmic function. This function is written as y = log⁡x. (There is no need to spend too much time on the notation. It will be solidified later.)

�Complete the discussion with problems 9–10. Ask students to show how they can use the graphs of f⁡(x) and f⁻¹⁡(x) to justify their answers. Emphasize problem 11, and ask students to make a general argument based on the three tasks and their knowledge of inverses so far. End the discussion by introducing a more formal definition of inverse functions as follows:�

In mathematics, an inverse function is a function that "reverses" or "undoes" another function. To describe this relationship in symbols, it can be said, The function g is the inverse of function f if and only if f⁡(a) = b and g⁡(b) = a. Using x and y, one would write f⁡(x) = y and g⁡(y) = x.�

NOTE�

Refer to your Teacher Edition: 5 Practices Charts book for preparation and guidance around in-class discussions.

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Lesson Summary

5 Minutes

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In this lesson,

LESSON SUMMARY

In this lesson, we modeled the inverse of an exponential function to determine its features.
We learned that this type of function is called a logarithmic function, which we will learn more about in Unit 2.
We also discussed a way to describe the input/relationship of inverse functions using mathematical notation.

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Algebra 2

Thank you