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 4.28
Textbook Question
Textbook QuestionGraph each function over a one-period interval.
y = -1 + csc x
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cosecant Function
The cosecant function, denoted as csc(x), is the reciprocal of the sine function. It is defined as csc(x) = 1/sin(x). The cosecant function has a range of all real numbers except for values between -1 and 1, and it is undefined wherever sin(x) equals zero. Understanding its behavior is crucial for graphing functions that involve csc(x).
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Vertical Shifts
Vertical shifts occur when a constant is added to or subtracted from a function. In the given function y = -1 + csc(x), the '-1' indicates a downward shift of the entire cosecant graph by one unit. This transformation affects the function's range and the position of its asymptotes, which are critical for accurately graphing the function.
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Phase Shifts
Periodicity of Trigonometric Functions
Trigonometric functions, including the cosecant function, are periodic, meaning they repeat their values in regular intervals. The period of csc(x) is 2π, which means the function will complete one full cycle over this interval. Recognizing the periodic nature of the function is essential for graphing it accurately over a one-period interval.
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