Unlocking Logarithmic Functions: Graphing and Transformations

PreAlgebra Grades High School 5:54 Video
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Lesson Description

Master the art of graphing logarithmic functions! This lesson delves into the relationship between exponential and logarithmic functions, explores transformations, and provides step-by-step examples for graphing complex logarithmic equations.

Video Resource

Logarithms (Logs) How to Graph

Mario's Math Tutoring

Duration: 5:54
Watch on YouTube

Key Concepts

  • Logarithmic Functions as Inverses of Exponential Functions
  • Transformations of Logarithmic Functions (Horizontal & Vertical Shifts)
  • Vertical Asymptotes and Domain Restrictions
  • Graphing logarithmic functions with transformations

Learning Objectives

  • Students will be able to identify the relationship between exponential and logarithmic functions.
  • Students will be able to graph logarithmic functions by applying transformations.
  • Students will be able to determine the domain and range of logarithmic functions.
  • Students will be able to identify vertical asymptotes of logarithmic functions.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of inverse functions and the graphical relationship between a function and its inverse (reflection over y = x). Briefly recap exponential functions and their graphs as a lead-in to logarithms.
  • Video Viewing & Note-Taking (10 mins)
    Play the Mario's Math Tutoring video "Logarithms (Logs) How to Graph." Instruct students to take notes on key concepts, examples, and the steps involved in graphing logarithmic functions with transformations.
  • Guided Practice: Example 1 (10 mins)
    Work through the first example from the video (y = log base 2 (x-1) + 3) step-by-step, emphasizing the effects of horizontal and vertical shifts on the parent function. Explain how to find the vertical asymptote and how it affects the domain.
  • Guided Practice: Example 2 (10 mins)
    Work through the second example from the video (y = log base 3 (x + 2) - 4), reinforcing the concepts of transformations and their impact on the graph. Encourage students to ask questions and participate in the problem-solving process.
  • Independent Practice (10 mins)
    Provide students with practice problems similar to the examples in the video. Have them graph logarithmic functions with different bases and transformations. Circulate to provide assistance and answer questions.

Interactive Exercises

  • Graphing Challenge
    Present students with a series of logarithmic functions with increasing complexity. Students compete to accurately graph the functions the fastest, reinforcing their understanding of transformations and key features.

Discussion Questions

  • How does changing the base of a logarithm affect its graph?
  • What is the relationship between the domain of a logarithmic function and its vertical asymptote?
  • How do transformations of logarithmic functions affect their domain and range?

Skills Developed

  • Graphing logarithmic functions
  • Identifying transformations of functions
  • Determining domain and range
  • Understanding inverse functions

Multiple Choice Questions

Question 1:

What is the inverse function of y = 2^x?

Correct Answer: y = log base 2 (x)

Question 2:

What transformation does the '+3' represent in the function y = log base 2 (x) + 3?

Correct Answer: Shift 3 units up

Question 3:

What transformation does the '(x-1)' represent in the function y = log base 2 (x-1)?

Correct Answer: Shift 1 unit to the right

Question 4:

Which of the following is true about the domain of y = log base b (x)?

Correct Answer: x > 0

Question 5:

What is a vertical asymptote?

Correct Answer: A line that the graph approaches but never touches.

Question 6:

How does the graph of y = log base b (x) behave as x approaches 0?

Correct Answer: Approaches negative infinity

Question 7:

Which of the following best describes the relationship between exponential and logarithmic functions with the same base?

Correct Answer: They are inverse functions

Question 8:

The graph of y = log base 4 (x + 5) has a vertical asymptote at:

Correct Answer: x = -5

Question 9:

What is the range of y = log base b (x), where b > 0 and b ≠ 1?

Correct Answer: All real numbers

Question 10:

Which transformation of y = log base 2 (x) results in a graph that is shifted downward by 2 units?

Correct Answer: y = log base 2 (x) - 2

Fill in the Blank Questions

Question 1:

The logarithmic function is the ________ of the exponential function.

Correct Answer: inverse

Question 2:

In the function y = log base b (x), 'b' is called the ________ of the logarithm.

Correct Answer: base

Question 3:

A vertical shift upwards occurs when a constant is ________ to the logarithmic function.

Correct Answer: added

Question 4:

The vertical asymptote of y = log base 5 (x - 3) is x = ________.

Correct Answer: 3

Question 5:

The domain of a logarithmic function is restricted to ________ numbers.

Correct Answer: positive

Question 6:

The graph of a logarithmic function gets closer and closer to the ________ asymptote.

Correct Answer: vertical

Question 7:

A horizontal shift left occurs when a number is ________ to x inside the log.

Correct Answer: added

Question 8:

The range of a logarithmic function is ________.

Correct Answer: all real numbers

Question 9:

The function y = log base 2 (x) passes through the point ( ________ , 0).

Correct Answer: 1

Question 10:

The log graph of y = log base b (x+a) shifts to the ________ by a units.

Correct Answer: left

Educational Standards

PC.F.1.2:[Objective] Sketch the graph of a function that models a relationship between two quantities, identifying key features. PC.F.1.4:[Objective] Describe end behavior, asymptotic behavior, and points of discontinuity. PC.F.1.5:[Objective] Determine if a function has an inverse. Algebraically and graphically find the inverse or define any restrictions on the domain that meet the requirement for invertibility, and find the inverse on the restricted domain.

Teaching Materials

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