Unveiling the Arccos Graph: Domain Restriction and Inverses

PreAlgebra Grades High School 3:13 Video
Print Lesson Plan Watch Video

Lesson Description

Explore the intricacies of graphing the arccos function, focusing on domain restriction, inverse functions, and graphical transformations. This lesson uses Mario's Math Tutoring video to provide a comprehensive understanding.

Video Resource

How to Graph Arccos (cosine inverse)

Mario's Math Tutoring

Duration: 3:13
Watch on YouTube

Key Concepts

  • Domain Restriction for Invertibility
  • Inverse Functions and Their Graphs
  • Reflection Over the Line y = x

Learning Objectives

  • Explain why the cosine function needs a restricted domain to have an inverse.
  • Graph the arccos function by switching x and y coordinates of key points on the restricted cosine function.
  • Describe the arccos graph as a reflection of the restricted cosine graph over the line y = x.

Educator Instructions

  • Introduction (5 mins)
    Begin by briefly reviewing the cosine function and its graph. Introduce the concept of inverse functions and why not all functions have inverses. Highlight the issue of the cosine graph failing the horizontal line test.
  • Domain Restriction (10 mins)
    Explain why the domain of the cosine function needs to be restricted to [0, π] to ensure its inverse is a function. Show the portion of the cosine graph that corresponds to this restricted domain.
  • Finding the Inverse Graphically (15 mins)
    Guide students through the process of finding the inverse function graphically. Emphasize switching the x and y coordinates of key points (0, 1), (π/2, 0), and (π, -1) on the restricted cosine graph to obtain points on the arccos graph. Plot these points and sketch the arccos graph.
  • Reflection Over y = x (5 mins)
    Demonstrate how the arccos graph is a reflection of the restricted cosine graph over the line y = x. Draw the line y = x and visually illustrate the reflection.
  • Conclusion (5 mins)
    Summarize the key concepts: domain restriction, inverse functions, and reflection over y = x. Briefly mention other inverse trigonometric functions (arcsin, arctan).

Interactive Exercises

  • Coordinate Switch
    Provide students with a set of coordinates from the restricted cosine function and ask them to switch the x and y values to find the corresponding coordinates on the arccos function. Then, have them plot these points.
  • Graphing Challenge
    Provide students with a blank coordinate plane and have them graph the restricted cosine function and the arccos function. Include the line y = x to demonstrate the reflection.

Discussion Questions

  • Why is it necessary to restrict the domain of the cosine function before finding its inverse?
  • How does the graph of a function relate to the graph of its inverse?
  • What are the key points on the arccos graph, and how do they relate to the restricted cosine graph?

Skills Developed

  • Graphical Representation of Functions
  • Understanding Inverse Functions
  • Application of Domain Restrictions
  • Critical Thinking

Multiple Choice Questions

Question 1:

Why is it necessary to restrict the domain of the cosine function when finding its inverse?

Correct Answer: To ensure the inverse is also a function.

Question 2:

What is the restricted domain of the cosine function used to define the arccos function?

Correct Answer: [0, π]

Question 3:

The graph of arccos(x) is a reflection of the restricted cosine graph over which line?

Correct Answer: y = x

Question 4:

What is the range of the arccos(x) function?

Correct Answer: [0, π]

Question 5:

Which of the following points lies on the graph of arccos(x)?

Correct Answer: (0, π/2)

Question 6:

The domain of arccos(x) is:

Correct Answer: [-1, 1]

Question 7:

If cos(π) = -1, then arccos(-1) equals:

Correct Answer: π

Question 8:

The arccos function is the inverse of which trigonometric function?

Correct Answer: Cosine

Question 9:

For what value of x does arccos(x) = 0?

Correct Answer: 1

Question 10:

What transformation is applied to the coordinates of the restricted cosine function to obtain the coordinates of the arccos function?

Correct Answer: Switching the x and y values

Fill in the Blank Questions

Question 1:

The domain of the cosine function is restricted to [0, ___] to define the arccos function.

Correct Answer: π

Question 2:

The graph of arccos(x) is a _______ of the restricted cosine function over the line y = x.

Correct Answer: reflection

Question 3:

Switching the x and y coordinates of a function produces its _______.

Correct Answer: inverse

Question 4:

The range of arccos(x) is [0, _______].

Correct Answer: π

Question 5:

arccos(1) = _______.

Correct Answer: 0

Question 6:

The arccos function answers the question: 'What _______ gives this value?'

Correct Answer: angle

Question 7:

A function must pass the _______ line test to have an inverse that is also a function.

Correct Answer: horizontal

Question 8:

The input values for the arccos function are the _______ values of the cosine function.

Correct Answer: x

Question 9:

arccos(0) = _______.

Correct Answer: π/2

Question 10:

To graph y = arccos(x), you can plot points by swapping the x and y coordinates of points on the restricted y = _______ graph.

Correct Answer: cos(x)

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