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 the Sine and Cosine Functions
Problem 17aBlitzer - 3rd Edition
Textbook Question
In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = sin(x − π)
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1
Identify the standard form of the sine function: .
Determine the amplitude by identifying the coefficient . In this case, , so the amplitude is 1.
Find the period of the function using the formula . Here, , so the period is .
Calculate the phase shift using . Here, and , so the phase shift is to the right.
Graph one period of the function by starting at the phase shift and plotting the sine wave over the interval .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Amplitude
Amplitude refers to the maximum height of a wave from its midline. In the context of the sine function, it indicates how far the graph reaches above and below the horizontal axis. For the function y = sin(x - π), the amplitude is 1, as the coefficient of the sine function is 1, meaning the graph oscillates between 1 and -1.
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Period
The period of a trigonometric function is the length of one complete cycle of the wave. For the sine function, the standard period is 2π. In the given function y = sin(x - π), there is no coefficient affecting the x variable, so the period remains 2π, indicating that the function will repeat its values every 2π units along the x-axis.
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Phase Shift
Phase shift refers to the horizontal displacement of a periodic function. It is determined by the value subtracted from the variable inside the function. In y = sin(x - π), the phase shift is π units to the right, as the function is shifted from the standard position of sin(x) to the right by π, affecting where the wave starts on the x-axis.
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Related Practice
Textbook Question
In Exercises 1–6, determine the amplitude of each function. Then graph the function and y = sin x in the same rectangular coordinate system for 0 ≤ x ≤ 2π.
y = 4 sin x
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