Textbook Question
Graph each function over the interval [-2π, 2π]. Give the amplitude. See Example 1.
y = 2 cos x
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 5
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 2 sin 2x
Verified step by step guidance1
Identify the general form of the sine function: \(y = A \sin(Bx - C) + D\), where \(A\) is the amplitude, \(B\) affects the period, \(C\) is the phase shift, and \(D\) is the vertical translation.
Compare the given function \(y = 2 \sin 2x\) to the general form. Here, \(A = 2\), \(B = 2\), \(C = 0\), and \(D = 0\).
Calculate the amplitude, which is the absolute value of \(A\): \(\text{Amplitude} = |2|\).
Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\), so substitute \(B = 2\) to find the period.
Determine the phase shift using \(\text{Phase shift} = \frac{C}{B}\). Since \(C = 0\), the phase shift is zero. The vertical translation \(D\) is also zero, meaning 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 of a Sine Function
Amplitude is the maximum value the sine function reaches from its midline. It is the absolute value of the coefficient before the sine term. For y = 2 sin 2x, the amplitude is 2, indicating the wave oscillates 2 units above and below the midline.
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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. It is calculated as 2π divided by the coefficient of x inside the sine function. For y = 2 sin 2x, the period is 2π/2 = π, meaning the wave repeats every π units.
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5:33Period of Sine and Cosine Functions
Vertical Translation and Phase Shift
Vertical translation shifts the graph up or down and is determined by any constant added outside the sine function; here, it is zero. Phase shift moves the graph left or right and depends on horizontal shifts inside the function's argument; since there is no added or subtracted value inside sin(2x), the phase shift is zero.
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6:31Phase Shifts
Related Practice
Textbook Question
Fill in the blank(s) to correctly complete each sentence.
The graph of y = 6 + 3 sin x is obtained by shifting the graph of y = 3 sin x ________ unit(s) __________ (up/down).
Textbook Question
Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = 2 sin ¼ x
Textbook Question
Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = π sin πx
Textbook Question
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = tan 3x
Textbook Question
Fill in the blank(s) to correctly complete each sentence.
The graph of y = -5 + 2 cos x is obtained by shifting the graph of y = 2 cos x ________ unit(s) __________ (up/down).
10
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