Mastering Tangent and Cotangent Graphs: A PreCalculus Exploration

PreAlgebra Grades High School 11:07 Video
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Lesson Description

This lesson explores the graphs of tangent and cotangent functions, focusing on their derivation from the unit circle, transformations (stretches, compressions, shifts, and reflections), and key characteristics like asymptotes and period. It also includes several examples of complex graphing problems.

Video Resource

Graphing Tan and Cot

Mario's Math Tutoring

Duration: 11:07
Watch on YouTube

Key Concepts

  • Tangent and Cotangent as Ratios on the Unit Circle
  • Asymptotes and Periodicity of Tangent and Cotangent
  • Transformations of Tangent and Cotangent Graphs (Vertical Stretch, Horizontal Compression, Phase Shift, Reflections)

Learning Objectives

  • Derive the graphs of tangent and cotangent functions from the unit circle.
  • Identify and graph transformations (stretches, compressions, shifts, and reflections) of tangent and cotangent functions.
  • Determine the period and asymptotes of transformed tangent and cotangent functions.
  • Apply transformation techniques to solve more complex graphing problems involving tangent and cotangent functions.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the definitions of tangent and cotangent in terms of sine and cosine and their relationship to the unit circle. Show the video from 0:01-0:50 as a refresher.
  • Graphing Parent Functions (10 mins)
    Guide students through graphing the parent functions y = tan(x) and y = cot(x). Emphasize the location of asymptotes and the period of each function. Show the video from 0:50 - 4:35 for a detailed walk through.
  • Transformations (15 mins)
    Explain how vertical stretches, horizontal compressions, horizontal shifts, and reflections affect the graphs of tangent and cotangent. Work through examples, modeling how to identify the transformations from the equation and apply them to the graph. Use video 4:35 - 6:39
  • Complex Graphing Problems (15 mins)
    Present more challenging problems involving multiple transformations. Encourage students to break down the problem into steps, first graphing the parent function, then applying transformations one at a time. Use video 6:39 - 9:15
  • Wrap-up and Q&A (5 mins)
    Summarize key concepts and address any remaining student questions.

Interactive Exercises

  • Graphing Transformations
    Students will use graphing software (e.g., Desmos, GeoGebra) to explore the effects of changing parameters in the equations of tangent and cotangent functions. They will be asked to predict the changes and then verify their predictions using the software.
  • Whiteboard Practice
    Present students with equations of transformed tangent and cotangent functions. Students work in small groups on whiteboards to sketch the graphs, identifying asymptotes, period, and key points.

Discussion Questions

  • How does the period of the tangent and cotangent functions differ from sine and cosine functions?
  • How do the asymptotes of the tangent and cotangent functions relate to the points on the unit circle where the functions are undefined?
  • Explain how reflections affect the general shape and direction of tangent and cotangent graphs.
  • Given the equation y=Atan(Bx +C) + D, describe what each variable is changing about the parent function

Skills Developed

  • Graphing Trigonometric Functions
  • Applying Transformations to Functions
  • Analyzing Function Equations
  • Problem-Solving

Multiple Choice Questions

Question 1:

What is the period of the parent function y = tan(x)?

Correct Answer: π

Question 2:

Which of the following transformations will stretch the graph of y = cot(x) vertically?

Correct Answer: y = 2cot(x)

Question 3:

Where do the asymptotes of the parent cotangent function occur?

Correct Answer: Multiples of π

Question 4:

Which transformation reflects the graph of y = tan(x) across the x-axis?

Correct Answer: y = -tan(x)

Question 5:

What transformation is represented by the equation y = tan(x - π/4)?

Correct Answer: Horizontal shift right π/4

Question 6:

What is the range of the parent tangent function?

Correct Answer: (-∞, ∞)

Question 7:

The graph of y = cot(Bx) has a period of π/3. What is the value of B?

Correct Answer: π/3

Question 8:

How does a vertical stretch affect the location of the asymptotes of a tangent function?

Correct Answer: Does not affect the location of the asymptotes

Question 9:

What is the domain of the cotangent function?

Correct Answer: All real numbers except multiples of π

Question 10:

Which of the following transformations compresses the graph of y = tan(x) horizontally?

Correct Answer: y = tan(2x)

Fill in the Blank Questions

Question 1:

The tangent function is defined as the ratio of the ______ coordinate to the ______ coordinate on the unit circle.

Correct Answer: y; x

Question 2:

The period of the cotangent function y = cot(x) is ______.

Correct Answer: π

Question 3:

A vertical stretch of the tangent graph y = tan(x) by a factor of 3 is represented by the equation y = ______.

Correct Answer: 3tan(x)

Question 4:

The asymptotes of the tangent function occur where cosine equals ______.

Correct Answer: zero

Question 5:

The transformation y = tan(x + π/2) shifts the graph of y = tan(x) to the ______ by π/2 units.

Correct Answer: left

Question 6:

Reflecting the cotangent function across the x-axis is represented by the equation y = ______.

Correct Answer: -cot(x)

Question 7:

If y = cot(Bx) has a period of π/4, then B = ______.

Correct Answer: 4

Question 8:

The parent tangent function passes through the origin which is the point ( ______ , ______ ).

Correct Answer: 0; 0

Question 9:

When the amplitude of the parent cotangent function is reflected, the function goes ______ as it progresses from left to right.

Correct Answer: up

Question 10:

The transformation y = tan(x) + 5 shifts the tangent graph 5 units ______.

Correct Answer: up

Educational Standards

PC.T.1.8:Graph all six trigonometric functions, identifying key features. PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation. PC.T.1.7:Apply the properties of a unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

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