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 4.12Lial - 12th Edition
Textbook Question
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -sin (x - 3π/4)
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Identify the standard form of the sine function: .
Compare the given function with the standard form to identify parameters: , , , and .
Determine the amplitude: The amplitude is the absolute value of , which is .
Calculate the period: The period of a sine function is given by . Since , the period is .
Identify the phase shift and vertical translation: The phase shift is to the right, and the vertical translation is , meaning there is no vertical shift.
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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 distance a wave reaches from its central axis or equilibrium position. For sine and cosine functions, it is determined by the coefficient in front of the function. In the case of y = -sin(x - 3π/4), the amplitude is 1, as the coefficient is -1, indicating the wave will oscillate 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 equation y = -sin(x - 3π/4), there is no coefficient affecting the x variable, so the period remains 2π, meaning the function will repeat every 2π units along the x-axis.
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Phase Shift
Phase shift refers to the horizontal displacement of a periodic function from its standard position. It is determined by the value subtracted from x in the function. In y = -sin(x - 3π/4), the phase shift is 3π/4 to the right, indicating that the entire sine wave is shifted 3π/4 units along 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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