Lesson 4

Lesson 4: Graphing Logarithmic Functions

 

Focus

 

The population of butterflies is dependent on the number of flower species the butterflies feed on. The relationship is a logarithmic function. As the number of flowers species, F, increases, so does the number of butterfly observations, B, but at a slow rate.

 

In the previous lesson you explored the meaning of logarithms and how they are evaluated. In this lesson you will explore graphing logarithmic functions and how transformations can be used to help sketch logarithmic functions. You will also look at how logarithmic functions can be used to solve problems.

Lesson Outcomes

 

At the end of this lesson you will be able to

• sketch the graph and identify the characteristics of a logarithmic function of the form y = logc x, c > 0 and c ≠ 1
• sketch the graph of a logarithmic function by applying transformations to the graph of y = logc x, c > 0 and c ≠ 1

Lesson Question

 

You will investigate the following question:

• How can logarithmic functions be graphed and described using transformations?

Assessment

 

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 4 Assignment (Download the Lesson 4 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Formula Sheet

Launch

 

Do you have the background knowledge and skills you need to complete this lesson successfully? Launch will help you find out.

 

Before beginning this lesson you should be able to

• identify functions
• identify inverse functions

Are You Ready?

 

Complete the following questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1. Examine each graph of y = f(x); then sketch the graph of its inverse. Determine whether or not the inverse is a function. Provide a reason for your answer.
   1. 
      
      Answer
   2. 
      
      Answer
2. Determine graphically whether each pair of functions are inverses of each other.
   1. f(x) = x − 4 and g(x) = x + 4 Answer
   2. f(x) = 3x + 5 and Answer
   3. f(x) = x − 7 and g(x) = 7 − x Answer

If you answered the Are You Ready? questions without difficulty, move to Discover.

 

If you found the Are You Ready? questions difficult, complete Refresher.

Refresher

 

Review inverse functions by watching “Introduction to Function Inverses.”

Go back to the Are You Ready? section and try the questions again. If you are still having difficulty, contact your teacher.

Discover

 

Try This 1

1. State the inverse of f(x) = 5x.
2. Complete the following tables of values.
   
   

   f(x) = 5x
   x	y
   −2
   −1
   0
   1
   2

   
   

   INVERSE OF f(x) = 5x
   f -1(x)
   x	y
   	−2
   	−1
   	0
   	1
   	2

3. Sketch the graphs of f(x) = 5x and the inverse, f −1(x).
4. Identify the following characteristics of the graph of f(x) = 5x and the inverse graph.
   
   

   Characteristics	f(x) = 5x	f -1(x)
   Domain
   Range
   x-intercept
   y-intercept
   Equation of Any Asymptotes

5. Open “Logarithmic Functions – Activity A.”
   

    

    
   
   Screenshot reprinted with permission of ExploreLearning
   
   
   Step 1: Change the a-value to 5.
   
   Step 2: Click on the box “Show associated exponential.”
   
   Step 3: Click on the box “Show line y = x.”
   
   Step 4: Compare your answers to Try This 1 to the graphs produced.
6. How are the x- and y-coordinates of the corresponding graphs related for the two functions?
7. Describe the transformation of the graph of f(x) = 5x to get the graph of the inverse.

Save your responses in your course folder.

 

Share 1

 

Based on your graphs created in Try This 1, discuss the relationship between the characteristics of the function f(x) = 5x and the inverse, f –1(x), with a partner or group.

 

If required, save a record of your discussion in your course folder.

Explore

 

In Try This 1 you graphed the exponential function f(x) = 5x and its inverse, f –1(x) = log5 x. The graph of f –1(x) = log5 x is the reflection of the graph of f(x) = 5x about the line y = x. The coordinates of the exponential function graph and its inverse are connected by the following map: (x, y) (y, x).

 

Self-Check 1

1. Use your knowledge of logarithmic functions to complete questions 1 to 5 in “Logarithmic Functions – Activity A.”
   
   
    
   
   Screenshot reprinted with permission of ExploreLearning

In Try This 2 you may have found that the parameters a, b, h, and k change the graph of a logarithmic function in the form in the following ways.

 

Parameter	Transformation	Example
a	There is a vertical stretch by a factor of |a| about the x-axis. When a < 0, there is a reflection in the x-axis.
b	There is a horizontal stretch by a factor of about the y-axis. When b < 0, there is a reflection in the y-axis.
h	There is a horizontal translation of h units left or right. The h-value affects the vertical asymptote.
k	There is a vertical translation of k units up or down.

 

You may have also noticed in Try This 2 that the parameters h and b affect the domain:

• The h-value translates the domain left or right.
• If the b-value is negative, the graph is reflected in the y-axis, and so the domain is reflected.

The range is unchanged and is all real numbers.

Try This 3

 

You will now explore graphing logarithmic functions using transformations.

1. In a table similar to the one shown, identify each parameter for the function y = log10(x − 3) + 4 when in the form
   
   

   Parameter	Value of Parameter	Description of Transformation
   a	1	no transformation
   b	1	no transformation
   h
   k

2. In your table, indicate the corresponding transformations for each parameter.
3. Sketch the function y = log10 x.
4. Apply the transformations described in the table to the sketch of y = log10x to sketch the graph of y = log10(x − 3) + 4.
5. Graph y = log10(x − 3) + 4 using technology, and then compare this to the sketch you created in question 4.

Save your responses in your course folder.

In Try This 3 you graphed a logarithmic function using transformations. You may have noticed that the function y = log10(x − 3) + 4 is the transformed graph of y = log10 x by a horizontal translation of right 3 units and a vertical translation of 4 units up.

 

Your graph may have looked like this one:

 

 

View Graphing Logarithmic Functions to see an example of how to graph a logarithmic function using transformations.

 

 

Self-Check 2

In Try This 3 you looked at graphing logarithmic functions from their equations. It is also possible to write the equation of a logarithmic function when given the graph of the function.

Try This 4

 

1. Determine algebraically the number of flower species if 100 butterflies are observed.
2. Determine algebraically the number of butterflies that should be observed if there are 20 flower species.
3. Use the graph of function F = −2.641 + 8.958 log10 B to check your solution to question 1. Explain why the graph shown in question 2 cannot be used to check your answer for question 2.
   
   
    
   
4. Graph, using technology, the function F = −2.641 + 8.958 log10 B to check your answer to question 2. Explain the process you used.

Save your responses in your course folder.

In Try This 4 you used a logarithmic function to determine the number of flowers and the number of butterflies observed in an area. You may have calculated that if there are 100 butterflies observed in an area, there would be approximately 15 flower species.

 

 

You may have found that when there are 20 flower species in an area, there should be approximately 337 butterflies observed.

 

 

You may have determined the number of butterflies observed by using a graph like this one:

 

 

The intersection point of the graph is approximately (337, 20), so approximately 337 butterflies would be observed when there are 20 flower species in an area.

 

Self-Check 4

Connect

 

Lesson 4 Assignment

Lesson 4 Summary

 

In this lesson you looked at graphing logarithmic functions and how transformations can be applied to logarithmic functions.

 

When a logarithmic function is in the form the parameters a, b, h, and k correspond to the following transformations of the graph of y = logc x:

 

Parameter	Value > 0	Value < 0
a	vertical stretch by a factor of |a|	vertical stretch by a factor of |a| and reflection in the x-axis
b	horizontal stretch of graph  by a factor of	horizontal stretch by a factor of  and reflection in the y-axis
h	translation of h units to the right	translation of |h| units to the left
k	translation of k units up	translation of |k| units down

 

In the next lesson you will continue to work with logarithms. You will explore how to work with multiple logarithms in an expression.