Lesson 2 Log In, Log Out Develop Understanding

Jump Start

Graph each pair of functions on the same graph.

1.

$y = x^{2}$ and $y = - {(x + 1)}^{2} + 3$

2.

$y = | x |$ and $y = | x - 3 | - 2$

Learning Focus

Identify and explain common features of the graphs of exponential functions.

Identify and explain common features of the graphs of logarithmic functions.

Transform the graph of an exponential or logarithmic function.

What are the features of the graphs of logarithmic functions?

How can we use the features of the parent function to transform exponential and logarithmic functions?

How does transforming the graphs of exponential and logarithmic functions compare to transforming quadratic functions?

Open Up the Math: Launch, Explore, Discuss

In Tracking the Tortoise, we built the graph of $y = \log_{2} x$ using the graph of $y = 2^{x}$ . In this task, we will be working with graphs again to make some generalizations about features of exponential functions and their related logarithmic functions.

1.

Let’s begin with comparing exponential functions. Graph the following two functions, being sure to include negative $x$ -values.

a.

$f (x) = 2^{x}$

b.

$g (x) = 3^{x}$

c.

What features, such as domain, range, end behavior, maximum, minimum, and $y$ -intercept, are the same in the two functions? What differences do you see?

d.

What features will be common for all exponential functions with a base greater than $1$ ? Explain why these features are the same.

2.

Now, let’s graph these two exponential functions:

a.

$f (x) = {(\frac{1}{2})}^{x}$

b.

$g (x) = {(\frac{1}{3})}^{x}$

c.

What features, such as domain, range, end behavior, maximum, minimum, and $y$ -intercept, are the same in the two functions? What differences do you see?

d.

What features will be common for all exponential functions with a base between $0$ and $1$ ? Explain why these features are the same.

e.

How is the graph of $f (x) = {(\frac{1}{2})}^{x}$ related to the graph of $g (x) = 2^{x}$ ? Why does this relationship exist?

3.

Construct a table of values and a graph for following two functions. Be sure to select at least two values in the interval: $0 < x < 1$ .

a.

$f (x) = \log_{2} x$

b.

$g (x) = \log_{3} x$

c.

What features, such as domain, range, end behavior, maximum, minimum, and $y$ -intercept, are the same in the two functions? What differences do you see?

d.

How is the graph of $f (x) = \log_{2} x$ related to the graph of $g (x) = 2^{x}$ ?

Pause and Reflect

In the past, we have transformed both quadratic and absolute value functions. Let’s try transforming exponential functions.

4.

First, consider the parent function $f (x) = 2^{x}$ . What equation would you write to shift the graph down $3 units$ ?

5.

Use technology to see if the equation you wrote in #4 works to shift the graph down $3 units$ . Revise your equation, if needed, and write it here.

6.

What equation would you write to shift the graph of $f (x) = 2^{x}$ to the left $3 units$ ?

7.

Use technology to see if the equation you wrote in #6 works to shift the graph left $3 units$ . Revise your equation, if needed, and write it here.

8.

Without using technology, graph this equation: $y = 1 + 2^{x + 2}$ .

9.

Write the equation for this graph:

10.

Here’s your last challenge: Write the equation of the function that transforms the graph of $f (x) = \log_{2} x$ up $2 units$ and to the right $1 unit$ .

Ready for More?

1.

Graph the function: $f (x) = \log_{\frac{1}{2}} x$ .

2.

What features of this function are the same as logarithmic functions with base $b > 1$ ?

3.

How does this function differ from logarithmic functions with base $b > 1$ ?

Takeaways

Features of logarithmic functions, $b > 1$ :

Quick Graphs for Exponential Functions:

Demonstrating with the function: $y = - 2 + 2^{x - 1}$ .

Adding Notation, Vocabulary, and Conventions

Vertical asymptote:

Vocabulary

horizontal shift
parent function
reflection
transformations on a function (non-rigid)
transformations on a function (rigid)
vertical asymptote
vertical shift
vertical stretch
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we graphed exponential and logarithmic functions by hand and used technology to determine common features in the graphs. We used technology to identify how the transformations appear in the equations of exponential functions. We examined graphs and wrote equations for the transformed functions, and we graphed transformed functions given the equations.

Retrieval

Find the answer to each logarithmic equation.

1.

$\log_{2} 16 =$

2.

$\log 100,000 =$

Rewrite each expression.

3.

${(n^{2})}^{5}$

4.

${(3 b^{- 4})}^{3}$