Mastering Tangent Graphs: A Simplified Approach

Algebra 2 Grades High School 2:33 Video
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Lesson Description

This lesson simplifies graphing tangent functions by focusing on period, vertical stretch, and horizontal shift. Learn to analyze and sketch tangent graphs with ease!

Video Resource

How to Graph Tangent (Simplified)

Mario's Math Tutoring

Duration: 2:33
Watch on YouTube

Key Concepts

  • Parent Tangent Function
  • Period of a Tangent Function
  • Vertical Stretch of a Tangent Function
  • Horizontal Shift (Phase Shift) of a Tangent Function
  • Asymptotes

Learning Objectives

  • Students will be able to determine the period of a tangent function given its equation.
  • Students will be able to identify and apply vertical stretches and horizontal shifts to the parent tangent function.
  • Students will be able to graph tangent functions accurately, including asymptotes and key points.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the unit circle and the definition of tangent as y/x. Introduce the concept of graphing tangent functions and highlight the challenges students often face. Show the video (0:00-0:29).
  • Understanding the Parent Tangent Function (10 mins)
    Discuss the parent tangent function, y = tan(x), focusing on its key characteristics: asymptotes at -π/2 and π/2, zero at x=0, and points at (-π/4, -1) and (π/4, 1). Explain how these values are derived from the unit circle. Show the video (0:29-1:18).
  • Period of Tangent Functions (10 mins)
    Introduce the formula for calculating the period of a tangent function: Period = π/B, where B is the coefficient of x. Work through Example 1 from the video: y = tan(1/2)x (1:18-2:56).
  • Vertical Stretch (10 mins)
    Explain how the coefficient 'A' in y = A tan(Bx) affects the vertical stretch of the graph. Work through Example 2 from the video: y = 2 tan(4x) (2:56-4:08).
  • Horizontal Shift (Phase Shift) (10 mins)
    Explain how the constant 'C' in y = tan(x - C) affects the horizontal shift of the graph. Work through Example 3 from the video: y = tan(x - π/4) (4:34-5:39).
  • Graphing Strategies and Review (5 mins)
    Review the key steps for graphing tangent functions: identify the period, vertical stretch, and horizontal shift; determine the location of asymptotes; plot key points; and sketch the curve. Emphasize the concave up/concave down nature of the graph around the origin. Show the video (4:08-4:34) again for review of curve sketching.

Interactive Exercises

  • Graphing Practice
    Provide students with a set of tangent function equations (e.g., y = 3 tan(2x), y = tan(x + π/2), y = 0.5 tan(x/3)). Have them graph these functions individually or in small groups, labeling the asymptotes and key points.
  • Equation Matching
    Create a matching activity where students match tangent function equations with their corresponding graphs.

Discussion Questions

  • How does the period of a tangent function relate to the coefficient of x?
  • What are the key differences between graphing tangent functions and graphing sine or cosine functions?
  • How do vertical stretches and horizontal shifts affect the asymptotes of a tangent function?

Skills Developed

  • Graphing trigonometric functions
  • Analyzing function transformations
  • Problem-solving using mathematical models

Multiple Choice Questions

Question 1:

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

Correct Answer: π

Question 2:

The period of y = tan(3x) is:

Correct Answer: π/3

Question 3:

What transformation does the coefficient '2' represent in the equation y = 2tan(x)?

Correct Answer: Vertical Stretch

Question 4:

The graph of y = tan(x - π/2) is the graph of y = tan(x) shifted:

Correct Answer: π/2 units to the right

Question 5:

Where do the asymptotes of the parent tangent function lie within the interval [-π, π]?

Correct Answer: -π/2 and π/2

Question 6:

What is the impact of a vertical stretch on the x-intercept of a tangent function?

Correct Answer: It does not change the x-intercept.

Question 7:

Which transformation alters the position of the vertical asymptotes in a tangent function?

Correct Answer: Horizontal shift

Question 8:

What is the value of tan(x) when x = 0?

Correct Answer: 0

Question 9:

Given the equation y = tan(Bx), if B > 1, the graph of the tangent function is:

Correct Answer: Compressed horizontally

Question 10:

Which of the following is true about the tangent function?

Correct Answer: It is undefined at certain points.

Fill in the Blank Questions

Question 1:

The tangent function, y = tan(x), is defined as y/x on the _________ circle.

Correct Answer: unit

Question 2:

The formula for finding the period of a tangent function is Period = π/_______.

Correct Answer: B

Question 3:

A vertical stretch of a tangent function is determined by the coefficient _______ in the equation y = A tan(Bx).

Correct Answer: A

Question 4:

A horizontal shift is also known as a _______ shift.

Correct Answer: phase

Question 5:

The parent tangent function has asymptotes at x = _______/2 and x = -_______/2.

Correct Answer: π

Question 6:

The tangent function is _______ at its asymptotes.

Correct Answer: undefined

Question 7:

In the equation y = tan(x + π/4), the graph is shifted _______ units to the left.

Correct Answer: π/4

Question 8:

When graphing the tangent function, the graph is concave _______ to the left of the origin.

Correct Answer: down

Question 9:

The x-intercept of the parent tangent function is at x = _______.

Correct Answer: 0

Question 10:

The range of the parent tangent function is all _______ numbers.

Correct Answer: real

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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