Textbook Question
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -½ cos 3x
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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 6
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).
Verified step by step guidance1
Identify the base function and the transformation: The base function is \(y = 2 \cos x\), and the given function is \(y = -5 + 2 \cos x\).
Recognize that adding or subtracting a constant outside the cosine function results in a vertical shift of the graph.
Since the function is \(y = 2 \cos x - 5\), this means the graph of \(y = 2 \cos x\) is shifted vertically by 5 units.
Determine the direction of the shift: because the constant is \(-5\), the graph shifts 5 units down.
Therefore, the graph of \(y = -5 + 2 \cos x\) is obtained by shifting the graph of \(y = 2 \cos x\) 5 units down.
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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 y = 2 cos x, adding or subtracting a constant outside the cosine function shifts the graph vertically by that constant amount.
Recommended video:
6:31
Phase Shifts
Amplitude and Vertical Translation
The amplitude of y = 2 cos x is 2, which affects the height of peaks and troughs. Adding -5 shifts the graph down by 5 units, changing the midline from y=0 to y=-5, but the amplitude remains the same.
Recommended video:
5:05Amplitude and Reflection of Sine and Cosine
Function Transformation Notation
In the function y = a cos x + d, 'd' represents the vertical shift. A positive 'd' shifts the graph up, while a negative 'd' shifts it down. Recognizing this helps identify how the graph moves relative to the base function y = a cos x.
Recommended video:
5:25Introduction to Transformations
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).
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Textbook Question
Fill in the blank(s) to correctly complete each sentence.
The graph of y = 3 + 5 cos (x + π/5) is obtained by shifting the graph of y = cos x horizontally ________ unit(s) to the __________, (right/left) stretching it vertically by a factor of ________, and then shifting it vertically ________ unit(s) __________. (up/down)
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Textbook Question
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 2 sin 2x
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Textbook Question
Fill in the blank(s) to correctly complete each sentence.
The graph of y = -3 sin x is obtained by stretching the graph of y = sin x by a factor of ________ and reflecting across the ________-axis.
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Textbook Question
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = tan 3x
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