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 23aBlitzer - 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 = 1/2 sin(x + π/2)
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Identify the standard form of the sine function: .
Determine the amplitude by identifying the coefficient . In this case, , so the amplitude is .
Find the period of the function using the formula . Here, , so the period is .
Calculate the phase shift using . With and , the phase shift is .
Graph one period of the function by starting at the phase shift and ending at , marking key points such as maximum, minimum, and intercepts based on the amplitude and period.
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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 sine functions, it is determined by the coefficient in front of the sine term. For the function y = 1/2 sin(x + π/2), the amplitude is 1/2, indicating that the wave oscillates between 1/2 and -1/2.
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Period
The period of a trigonometric function is the length of one complete cycle of the wave. For sine functions, the standard period is 2π. However, if the function includes a coefficient affecting the x variable, the period is calculated as 2π divided by that coefficient. In this case, the period remains 2π since there is no coefficient affecting x.
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Phase Shift
Phase shift refers to the horizontal displacement of a wave from its standard position. It is determined by the value added or subtracted from the x variable inside the function. For y = 1/2 sin(x + π/2), the phase shift is -π/2, meaning the graph is shifted to the left by π/2 units.
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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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