Textbook Question
Fill in the blank(s) to correctly complete each sentence.
The graph of y = cos (x - π/6) is obtained by shifting the graph of y = cos x ______ unit(s) to the ________ (right/left).
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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 1
Fill in the blank(s) to correctly complete each sentence.
The graph of y = sin (x + π/4) is obtained by shifting the graph of y = sin x ______ unit(s) to the ________ (right/left).
Verified step by step guidance1
Recall that the function y = sin(x + c) represents a horizontal shift of the basic sine function y = sin x, where c is a constant.
If the function is y = sin(x + \(\frac{\pi}{4}\)), the graph is shifted horizontally by \(\frac{\pi}{4}\) units.
Since the argument inside the sine function is (x + \(\frac{\pi}{4}\)), this corresponds to a shift to the left by \(\frac{\pi}{4}\) units (because adding inside the function shifts the graph left).
Therefore, the graph of y = sin(x + \(\frac{\pi}{4}\)) is obtained by shifting the graph of y = sin x \(\frac{\pi}{4}\) unit(s) to the left.
This is a standard property of function transformations: y = sin(x + c) shifts the graph left by c units, and y = sin(x - c) shifts it right by c units.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Phase Shift in Trigonometric Functions
Phase shift refers to the horizontal translation of a trigonometric graph caused by adding or subtracting a constant inside the function's argument. For y = sin(x + π/4), the graph shifts horizontally by π/4 units. The sign inside the function determines the direction of the shift.
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6:31
Phase Shifts
Graph of the Sine Function
The sine function y = sin x is periodic with period 2π and oscillates between -1 and 1. Understanding its basic shape and key points helps in visualizing transformations like shifts, stretches, and reflections applied to the graph.
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5:53Graph of Sine and Cosine Function
Effect of Adding a Positive Constant Inside the Function Argument
Adding a positive constant inside the argument of sine, as in sin(x + c), shifts the graph to the left by c units. This is because the input values that produce the same output occur earlier on the x-axis, effectively moving the graph leftward.
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3:30Example 3
Related Practice
Textbook Question
An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follows, where t is time in seconds.
𝒮(t) = 5 cos 2t
What is the amplitude of this motion?
Textbook Question
Fill in the blank(s) to correctly complete each sentence.
The graph of y = 4 sin x is obtained by stretching the graph of y = sin x vertically by a factor of ________.
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Textbook Question
An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follows, where t is time in seconds.
𝒮(t) = 5 cos 2t
What is the period of this motion?
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