Graphing Tangent and Cotangent Functions with Transformations

Algebra 2 Grades High School 33:48 Video
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Lesson Description

Learn how to graph tangent and cotangent functions, including transformations like stretching, shifting, and reflecting. This lesson explores two methods: a shifted origin approach and a table-based transformation method.

Video Resource

How to Graph Tan and Cot (2 Methods)

Mario's Math Tutoring

Duration: 33:48
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Key Concepts

  • Parent functions of tangent and cotangent
  • Vertical asymptotes
  • Period of tangent and cotangent functions
  • Transformations: vertical stretch/shrink, horizontal stretch/shrink, horizontal shift, vertical shift, reflection

Learning Objectives

  • Students will be able to identify the parent functions of tangent and cotangent.
  • Students will be able to graph tangent and cotangent functions using two different methods.
  • Students will be able to determine the period, vertical asymptotes, domain, and range of transformed tangent and cotangent functions.
  • Students will be able to apply transformations (stretching, shifting, reflection) to tangent and cotangent functions.

Educator Instructions

  • Introduction to Tangent and Cotangent (5 mins)
    Review the unit circle and the definitions of tangent (y/x) and cotangent (x/y) in relation to the unit circle. Briefly discuss why tangent and cotangent have vertical asymptotes.
  • Graphing the Parent Tangent Function (10 mins)
    Using the unit circle values, plot key points for the tangent function from -π/2 to π/2. Identify the vertical asymptotes at -π/2 and π/2. Sketch the parent tangent function, emphasizing its shape and direction.
  • Graphing the Parent Cotangent Function (10 mins)
    Using the unit circle values, plot key points for the cotangent function from 0 to π. Identify the vertical asymptotes at 0 and π. Sketch the parent cotangent function, emphasizing its shape and direction.
  • Method 1: Shifted Origin Approach (15 mins)
    Explain how to determine the period of transformed tangent and cotangent functions (π/B). Show how to find the new 'origin' after horizontal and vertical shifts. Demonstrate graphing using the shifted origin and considering vertical stretches/shrinks and reflections.
  • Method 2: Table Transformation Approach (15 mins)
    Provide a table of key values for the parent tangent and cotangent functions. Explain how each transformation (A, B, H, K) affects the x and y values in the table. Demonstrate plotting the transformed points to graph the function.
  • Practice Problems (15 mins)
    Work through several examples of graphing transformed tangent and cotangent functions using both methods. Encourage students to try both methods and compare their results.
  • Domain and Range (5 mins)
    Discuss how transformations affect the domain and range of tangent and cotangent functions. Emphasize the importance of identifying vertical asymptotes to determine the domain.

Interactive Exercises

  • Transformation Challenge
    Provide students with equations of transformed tangent and cotangent functions. Ask them to predict the transformations and sketch the graph based on their predictions. Then, have them use graphing software to verify their answers.
  • Graphing Relay Race
    Divide students into teams. Provide each team with an equation of a transformed tangent or cotangent function. Each team member graphs one key feature of the function (asymptote, key point, etc.) on a shared graph. The first team to accurately graph the entire function wins.

Discussion Questions

  • How do the definitions of tangent and cotangent on the unit circle explain their behavior (increasing/decreasing, asymptotes)?
  • What are the advantages and disadvantages of each graphing method (shifted origin vs. table transformation)?
  • How does the 'B' value affect the period and horizontal stretch/shrink of tangent and cotangent functions?
  • How do vertical shifts affect the range of the tangent and cotangent functions?
  • Why are vertical asymptotes so important when graphing tangent and cotangent?

Skills Developed

  • Graphing trigonometric functions
  • Applying transformations to functions
  • Analyzing functions (domain, range, asymptotes)
  • Problem-solving
  • Analytical Thinking

Multiple Choice Questions

Question 1:

What is the period of the parent tangent function?

Correct Answer: π

Question 2:

Which transformation is represented by the 'A' value in y = A tan(Bx + H) + K?

Correct Answer: Vertical stretch/shrink

Question 3:

What is the vertical asymptote of the parent cotangent function on the interval [0, pi]?

Correct Answer: x = 0

Question 4:

A vertical shift is represented by which value in the tangent/cotangent equation?

Correct Answer: K

Question 5:

What is the range of the parent tangent function?

Correct Answer: (-∞, ∞)

Question 6:

The 'H' value in y = A tan(Bx + H) + K represents what type of transformation?

Correct Answer: Horizontal Shift

Question 7:

What is the domain restriction of a graph containing vertical asymptotes?

Correct Answer: x cannot equal the value where the vertical asymptote is located

Question 8:

Which of the following is the value of tangent at 0?

Correct Answer: 0

Question 9:

Which function goes down to the right on the coordinate plane?

Correct Answer: Cotangent

Question 10:

A reflection over the x axis is the result of which value being negative in the equation: y = A tan(Bx + H) + K?

Correct Answer: A

Fill in the Blank Questions

Question 1:

The tangent function is defined as y/x, while the cotangent function is defined as _____/y.

Correct Answer: x

Question 2:

Vertical asymptotes occur where the denominator of a rational function is equal to ______.

Correct Answer: zero

Question 3:

The period of y = tan(Bx) is calculated by π/____.

Correct Answer: B

Question 4:

A vertical stretch of a tangent or cotangent function occurs when the 'A' value is greater than ______.

Correct Answer: 1

Question 5:

The graph of cotangent goes ______ to the right.

Correct Answer: down

Question 6:

A horizontal shift to the left occurs when H is _______.

Correct Answer: positive

Question 7:

The B value has a ________ effect on the x values when plotting.

Correct Answer: reciprocal

Question 8:

When using the table transformation approach, do transformations from _______ to _______.

Correct Answer: left, right

Question 9:

Vertical asymptotes are used to determine the __________ of the function.

Correct Answer: domain

Question 10:

Half of the tangent function is to the _______ of the y axis.

Correct Answer: left

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.1.6:Graph a rational function and identify the domain (including holes), range, x- and y-intercepts, vertical and horizontal asymptotes, using various methods and tools that may include a graphing calculator or other appropriate technology (excluding slant or oblique asymptotes).

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