Table of contents
- 0. Review of College Algebra4h 43m
- 1. Measuring Angles39m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
4. Graphing Trigonometric Functions
Graphs of Tangent and Cotangent Functions
Problem 4.9b
Textbook Question
Match each function with its graph in choices A–F.
y = tan (x - π )
A. <IMAGE> B. <IMAGE> C. <IMAGE>
D. <IMAGE> E. <IMAGE> F. <IMAGE>
![](/channels/images/assetPage/verifiedSolution.png)
1
Identify the basic form of the tangent function, \( y = \tan(x) \), which has vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \).
Recognize that the given function is \( y = \tan(x - \pi) \), which represents a horizontal shift of the basic tangent function.
Understand that a horizontal shift of \( \pi \) units to the right means the graph of \( y = \tan(x) \) is moved \( \pi \) units to the right.
Determine the new positions of the vertical asymptotes, which will now be at \( x = \frac{\pi}{2} + n\pi + \pi = \frac{3\pi}{2} + n\pi \).
Match the transformed graph with the correct image choice by looking for the graph with vertical asymptotes at \( x = \frac{3\pi}{2} + n\pi \) and the same periodicity as the tangent function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Function Properties
The tangent function, defined as tan(x) = sin(x)/cos(x), has specific properties including periodicity, asymptotes, and behavior around its vertical asymptotes. It is periodic with a period of π, meaning it repeats every π units. Understanding these properties is crucial for identifying the correct graph, especially how the function behaves as it approaches its asymptotes.
Recommended video:
Introduction to Tangent Graph
Phase Shift
A phase shift occurs when a function is horizontally translated along the x-axis. In the function y = tan(x - π), the graph of the tangent function is shifted π units to the right. Recognizing this shift is essential for accurately matching the function to its corresponding graph, as it alters the location of the function's key features, such as its asymptotes and intercepts.
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Phase Shifts
Graphing Trigonometric Functions
Graphing trigonometric functions involves plotting key points, identifying asymptotes, and understanding the function's periodic nature. For the tangent function, it is important to note where the function is undefined (asymptotes) and how the graph behaves between these points. Familiarity with the general shape of the tangent graph helps in selecting the correct graph from the given options.
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Introduction to Trigonometric Functions
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