Textbook Question
Identify the circular function that satisfies each description.
period is π; function is decreasing on the interval (0, π)
Ch. 4 - Graphs of the Circular Functions
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 5, Problem 21
Match each function in Column I with the appropriate description in Column II.
I
y = -4 sin(3x - 2)
II
A. amplitude = 2, period = π/2, phase shift = ¾
B. amplitude = 3, period = π, phase shift = 2
C. amplitude = 4, period = 2π/3, phase shift = ⅔
D. amplitude = 2, period = 2π/3, phase shift = 4⁄3
Verified step by step guidance1
Identify the general form of the sine function: \(y = A \sin(Bx - C)\), where \(A\) is the amplitude, \(B\) affects the period, and \(C\) affects the phase shift.
Determine the amplitude by taking the absolute value of the coefficient in front of the sine function: \(|A| = |-4| = 4\).
Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Here, \(B = 3\), so the period is \(\frac{2\pi}{3}\).
Find the phase shift using the formula \(\text{Phase shift} = \frac{C}{B}\). Since the function is \(y = -4 \sin(3x - 2)\), rewrite the inside as \(3x - 2 = 3(x - \frac{2}{3})\), so the phase shift is \(\frac{2}{3}\).
Match the calculated values: amplitude = 4, period = \(\frac{2\pi}{3}\), phase shift = \(\frac{2}{3}\) with the options in Column II to find the correct description.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Amplitude of a Sine Function
Amplitude is the absolute value of the coefficient before the sine function and represents the maximum vertical displacement from the midline. For y = -4 sin(3x - 2), the amplitude is |−4| = 4, indicating the wave oscillates 4 units above and below the center line.
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Amplitude and Reflection of Sine and Cosine
Period of a Sine Function
The period is the length of one complete cycle of the sine wave and is calculated by dividing 2π by the coefficient of x inside the function. Here, the period is 2π/3 because the coefficient of x is 3, meaning the function repeats every 2π/3 units.
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5:33Period of Sine and Cosine Functions
Phase Shift of a Sine Function
Phase shift is the horizontal translation of the sine curve and is found by solving the inside of the function's argument for zero: 3x - 2 = 0, so x = 2/3. This means the graph shifts right by 2/3 units, adjusting where the wave starts along the x-axis.
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6:31Phase Shifts
Related Practice
Textbook Question
Match each function in Column I with the appropriate description in Column II.
I
y = 3 sin(2x - 4)
II
A. amplitude = 2, period = π/2, phase shift = ¾
B. amplitude = 3, period = π, phase shift = 2
C. amplitude = 4, period = 2π/3, phase shift = ⅔
D. amplitude = 2, period = 2π/3, phase shift = 4⁄3
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Textbook Question
Graph each function over a one-period interval.
y = ½ cot (4x)
Textbook Question
Graph each function over a one-period interval.
y = -2 tan (¼ x)
Textbook Question
Graph each function over a one-period interval.
y = cot (3x)
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Textbook Question
Graph each function over a two-period interval.
y = tan(2x - π)
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