Textbook Question
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -½ cos 3x
3
views
Ch. 4 - Graphs of the Circular Functions
Lial12th EditionTrigonometryISBN: 9780136552161
Not the one you use?Change textbookSelect a different textbook
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Was this helpful?
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).
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)
Textbook Question
Fill in the blank(s) to correctly complete each sentence.
The graph of y = -2 + 3 cos (x - π/6) 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)
Textbook Question
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 2 sin 2x
1
views
Textbook Question
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = tan 3x