Unveiling the Arctangent: Graphing the Inverse Tangent Function

PreAlgebra Grades High School 3:51 Video
Print Lesson Plan Watch Video

Lesson Description

This lesson explores the arctangent function, its graph, and its relationship to the tangent function. Students will learn how to graph the arctangent function by understanding domain restrictions, interchanging x and y values, and transforming asymptotes.

Video Resource

How to Graph Arctan (tangent inverse)

Mario's Math Tutoring

Duration: 3:51
Watch on YouTube

Key Concepts

  • Tangent Function
  • Inverse Functions
  • Domain Restriction
  • Asymptotes
  • Graphing Techniques

Learning Objectives

  • Students will be able to identify the domain restriction necessary for the tangent function to have an inverse.
  • Students will be able to graph the arctangent function by interchanging x and y values of the restricted tangent function.
  • Students will be able to transform vertical asymptotes of the tangent function into horizontal asymptotes of the arctangent function.
  • Students will be able to identify key features of the arctangent graph, including asymptotes, intercepts, and general shape.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the concept of inverse functions and the horizontal line test. Show the graph of the tangent function and discuss why it doesn't have an inverse over its entire domain.
  • Restricting the Domain (5 mins)
    Explain why mathematicians restrict the domain of the tangent function to (-π/2, π/2) to ensure it passes the horizontal line test and has an inverse. Show the restricted tangent graph.
  • Graphing the Arctangent (15 mins)
    Guide students through the process of graphing the arctangent function by interchanging x and y values of key points on the restricted tangent function graph (e.g., (-π/4, -1), (0, 0), (π/4, 1)). Emphasize the reflection of the graph over the line y = x.
  • Asymptotes (10 mins)
    Explain how the vertical asymptotes of the tangent function at x = -π/2 and x = π/2 become horizontal asymptotes of the arctangent function at y = -π/2 and y = π/2. Demonstrate this graphically.
  • Practice and Examples (10 mins)
    Work through additional examples of graphing the arctangent function, emphasizing the key steps. Address any student questions.

Interactive Exercises

  • Graphing Challenge
    Provide students with a blank coordinate plane and have them graph the arctangent function, labeling key points and asymptotes. They can compare their graphs with a provided solution.
  • Point Swap Activity
    Give students a list of coordinates from the restricted tangent function and have them swap the x and y values to find the corresponding points on the arctangent function. Then plot these points.

Discussion Questions

  • Why is it necessary to restrict the domain of the tangent function before finding its inverse?
  • How do the asymptotes of the tangent function relate to the asymptotes of the arctangent function?
  • What are some real-world applications of the arctangent function?

Skills Developed

  • Graphing Functions
  • Understanding Inverse Functions
  • Analytical Thinking
  • Problem-Solving

Multiple Choice Questions

Question 1:

What is the restricted domain of the tangent function used to define the arctangent function?

Correct Answer: (-π/2, π/2)

Question 2:

The vertical asymptotes of the tangent function become what feature of the arctangent function?

Correct Answer: Horizontal Asymptotes

Question 3:

Which of the following points lies on the graph of the arctangent function?

Correct Answer: (1, π/4)

Question 4:

What is the range of the arctangent function?

Correct Answer: (-π/2, π/2)

Question 5:

The graph of arctangent is a reflection of the restricted tangent graph over which line?

Correct Answer: y = x

Question 6:

What is the value of arctan(0)?

Correct Answer: 0

Question 7:

What is the equation of the horizontal asymptote of arctan(x) as x approaches positive infinity?

Correct Answer: y = π/2

Question 8:

If tan(θ) = x, then arctan(x) = ?

Correct Answer: x

Question 9:

The arctangent function is also written as:

Correct Answer: tan⁻¹(x)

Question 10:

Which of the following statements is true regarding the domain of the standard arctangent function?

Correct Answer: (-∞, ∞)

Fill in the Blank Questions

Question 1:

The inverse of the tangent function is called the ________ function.

Correct Answer: arctangent

Question 2:

The domain of arctan(x) is all ________ numbers.

Correct Answer: real

Question 3:

The range of arctan(x) is (_, _).

Correct Answer: (-π/2, π/2)

Question 4:

The horizontal asymptotes of the arctangent function are y = π/2 and y = _______.

Correct Answer: -π/2

Question 5:

To graph the inverse of a function, you interchange the ______ and ______ coordinates.

Correct Answer: x, y

Question 6:

The domain of the tangent function is restricted to (-π/2, π/2) to ensure it passes the ________ line test.

Correct Answer: horizontal

Question 7:

arctan(1) = _______.

Correct Answer: π/4

Question 8:

As x approaches negative infinity, arctan(x) approaches _______.

Correct Answer: -π/2

Question 9:

The arctangent function is the ________ of the tangent function.

Correct Answer: inverse

Question 10:

The graph of arctan(x) passes through the point (0, _______).

Correct Answer: 0

Educational Standards

PC.F.1.5: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. PC.T.1.8:Graph all six trigonometric functions, identifying key features. PC.F.1.2:Sketch the graph of a function that models a relationship between two quantities, identifying key features.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans