Mastering Logarithmic Functions: Transformations and Graphs

Algebra 2 Grades High School 9:40 Video
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Lesson Description

Learn to graph logarithmic functions with transformations, understand vertical asymptotes, and determine domain and range. This lesson uses examples to build skills in transforming logarithmic to exponential form and translating functions.

Video Resource

Graphing Logarithmic Functions with Transformations

Mario's Math Tutoring

Duration: 9:40
Watch on YouTube

Key Concepts

  • Logarithmic Functions as Inverses of Exponential Functions
  • Transformations of Logarithmic Functions (Horizontal and Vertical Shifts)
  • Vertical Asymptotes, Domain, and Range of Logarithmic Functions
  • Converting between logarithmic and exponential forms.

Learning Objectives

  • Students will be able to transform logarithmic functions from logarithmic to exponential form.
  • Students will be able to identify and apply transformations (horizontal and vertical shifts) to graph logarithmic functions.
  • Students will be able to determine the vertical asymptote, domain, and range of a transformed logarithmic function.
  • Students will be able to graph logarithmic functions with bases between 0 and 1.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the relationship between exponential and logarithmic functions as inverses. Briefly discuss the graph of a basic exponential function and its reflection over y = x to obtain the basic logarithmic function. Show the video (Mario's Math Tutoring - Graphing Logarithmic Functions with Transformations).
  • Understanding Transformations (10 mins)
    Explain the general form y = log_b(x - h) + k. Discuss the effects of 'h' (horizontal shift) and 'k' (vertical shift) on the graph. Emphasize that 'h' has the opposite effect (e.g., x - 1 shifts the graph right by 1). Review how to find the vertical asymptote (x = h).
  • Converting to Exponential Form (10 mins)
    Review how to convert logarithmic equations to exponential equations (b^y = x). Practice converting several examples. Highlight that this conversion can simplify the process of graphing.
  • Graphing with Transformations (15 mins)
    Work through examples of graphing transformed logarithmic functions. For each example: 1. Identify h and k, and the vertical asymptote. 2. Convert to exponential form. 3. Create a table of values, substituting for y. 4. Plot the points, starting from the 'new origin' (h, k). 5. Sketch the graph, approaching the vertical asymptote.
  • Domain and Range (5 mins)
    Explain how the vertical asymptote determines the domain (x > h). Emphasize that the range of all transformed logarithmic functions is all real numbers.
  • Practice and Examples (10 mins)
    Present additional practice problems for students to solve independently or in small groups. Circulate to provide assistance and answer questions.

Interactive Exercises

  • Graphing Challenge
    Provide students with a worksheet containing several logarithmic functions with various transformations. Students work individually or in pairs to graph the functions and identify their key features (asymptote, domain, range).
  • Transformation Matching Game
    Create cards with logarithmic functions and corresponding cards with their transformed graphs. Students match the functions to their graphs.

Discussion Questions

  • How does the base of the logarithm affect the shape of the graph?
  • How does changing 'h' affect the vertical asymptote and the domain of the function?
  • Why is the range of a transformed logarithmic function always all real numbers?
  • Explain the relationship between a logarithmic function and it's inverse exponential function.

Skills Developed

  • Graphing logarithmic functions
  • Converting between logarithmic and exponential forms
  • Identifying and applying transformations
  • Determining domain and range

Multiple Choice Questions

Question 1:

What is the vertical asymptote of the function y = log₂(x - 3) + 1?

Correct Answer: x = 3

Question 2:

Which transformation shifts the graph of y = log(x) to the left by 2 units?

Correct Answer: y = log(x + 2)

Question 3:

What is the domain of the function y = log₅(x + 4)?

Correct Answer: x > -4

Question 4:

Which of the following is the exponential form of log₃(9) = 2?

Correct Answer: 3² = 9

Question 5:

The range of the function y = log(x) + k is:

Correct Answer: All real numbers

Question 6:

The graph of y = log_b(x) is reflected over the line _______ to obtain the graph of y=b^x?

Correct Answer: y = x

Question 7:

Which transformations are applied to y = log(x) to obtain the graph of y = -log(x+1) - 3?

Correct Answer: Reflection over the x-axis, shifts left 1 and down 3.

Question 8:

What is the value of 'h' in the general form y = log_b(x - h) + k if the graph has a vertical asymptote at x = -5?

Correct Answer: -5

Question 9:

What is the base of the common logarithm?

Correct Answer: 10

Question 10:

How does a vertical shift of +5 affect the domain of y = log_b(x)?

Correct Answer: The domain remains unchanged

Fill in the Blank Questions

Question 1:

The inverse of an exponential function is a ________ function.

Correct Answer: logarithmic

Question 2:

The line that a logarithmic function approaches but never touches is called the ________.

Correct Answer: asymptote

Question 3:

In the function y = log_b(x - h) + k, the variable 'h' represents a ________ shift.

Correct Answer: horizontal

Question 4:

To convert log₄(16) = 2 to exponential form, we write ________ = 16.

Correct Answer: 4^2

Question 5:

The ________ of a logarithmic function is always all real numbers.

Correct Answer: range

Question 6:

In the function y = log_b(x - h) + k, the ________ is x = h.

Correct Answer: vertical asymptote

Question 7:

When graphing y = log(x + 3), the graph shifts ________ by 3 units.

Correct Answer: left

Question 8:

The expression b to the power of y equals x can be written in logarithmic form as ________

Correct Answer: log_b(x)=y

Question 9:

Multiplying a log function by -1 such as y=-log(x) results in a reflection over the ________ axis.

Correct Answer: x

Question 10:

If a base b for log_b(x) is between 0 and 1, the graph will look different than a graph where the base is ________.

Correct Answer: greater than 1

Educational Standards

A2.F.1.2:Identify the parent forms of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [𝑓(𝑥 + 𝑐), 𝑓(𝑥) + 𝑐, 𝑓(𝑐𝑥), and 𝑐𝑓(𝑥)] algebraically and graphically. A2.F.1.4:Graph exponential and logarithmic functions. Identify the domain, range, asymptotes, and x- and y-intercepts using various methods and tools that may include calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. A2.A.1.6:Solve common and natural logarithmic equations using the properties of logarithms.

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