In Exercises 75–78, graph one period of each function. y = |2 cos x/2|
Table of contents
- 0. Review of College Algebra4h 45m
- 1. Measuring Angles40m
- 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 8
Textbook Question
Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −3 cos (x + π)
Verified step by step guidance1
Identify the general form of the cosine function: \(y = A \cos(B(x - C))\), where \(A\) is the amplitude, \(\frac{2\pi}{B}\) is the period, and \(C\) is the phase shift.
Rewrite the given function \(y = -3 \cos(x + \pi)\) in the form \(y = A \cos(B(x - C))\). Notice that \(x + \pi\) can be written as \(x - (-\pi)\), so \(C = -\pi\).
Determine the amplitude \(A\) by taking the absolute value of the coefficient in front of the cosine: \(A = | -3 | = 3\).
Find the period using the formula \(\text{Period} = \frac{2\pi}{B}\). Here, \(B\) is the coefficient of \(x\), which is 1, so the period is \(2\pi\).
Identify the phase shift \(C\), which is \(-\pi\). This means the graph is shifted \(\pi\) units to the left. Use this information to sketch one full period of the function starting from \(x = -\pi\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Amplitude of a Trigonometric Function
Amplitude is the maximum absolute value of the function's output, representing the height from the midline to the peak. For y = -3 cos(x + π), the amplitude is |−3| = 3, indicating the graph oscillates 3 units above and below its midline.
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Period of a Cosine Function
The period is the length of one complete cycle of the function. For y = cos(bx), the period is calculated as 2π/|b|. In y = −3 cos(x + π), since b = 1, the period is 2π, meaning the function repeats every 2π units along the x-axis.
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Period of Sine and Cosine Functions
Phase Shift of a Trigonometric Function
Phase shift is the horizontal translation of the graph, determined by solving inside the function for zero: x + π = 0 gives x = −π. This means the graph shifts π units to the left, affecting where the cosine wave starts its cycle.
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