Textbook Question
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -½ cos 3x
3
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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
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).
Verified step by step guidance1
Identify the base function and the transformation applied. The base function here is \(y = 3 \sin x\), and the transformed function is \(y = 6 + 3 \sin x\).
Recognize that adding a constant outside the sine function results in a vertical shift of the graph.
The constant added is 6, so the graph of \(y = 3 \sin x\) is shifted vertically by 6 units.
Since the constant is positive, the shift is upwards.
Therefore, the graph of \(y = 6 + 3 \sin x\) is obtained by shifting the graph of \(y = 3 \sin x\) 6 units up.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertical Shifts in Trigonometric Graphs
A vertical shift moves the entire graph of a function up or down without changing its shape. For a function y = f(x) + k, the graph shifts vertically by k units; if k is positive, the shift is upward, and if negative, downward.
Recommended video:
6:31
Phase Shifts
Amplitude of a Sine Function
The amplitude of y = a sin x is the absolute value of a, representing the maximum distance from the midline to the peak or trough. In y = 3 sin x, the amplitude is 3, which remains unchanged by vertical shifts.
Recommended video:
5:05Amplitude and Reflection of Sine and Cosine
Midline of a Trigonometric Function
The midline is the horizontal line around which the sine or cosine graph oscillates. For y = 3 sin x, the midline is y = 0; adding 6 shifts the midline to y = 6, indicating a vertical shift upward by 6 units.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Graph each function over the interval [-2π, 2π]. Give the amplitude. See Example 1.
y = 2 cos x
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
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 2 sin 2x
1
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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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