Textbook Question
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).
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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 3
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 ________.
Verified step by step guidance1
Identify the base function and the transformed function. Here, the base function is \(y = \sin x\) and the transformed function is \(y = 4 \sin x\).
Recall that the coefficient in front of the sine function affects the amplitude, which is the vertical stretch or compression of the graph.
The amplitude of \(y = \sin x\) is 1, since the sine function oscillates between -1 and 1.
The amplitude of \(y = 4 \sin x\) is 4, meaning the graph is stretched vertically by a factor equal to the amplitude.
Therefore, the vertical stretch factor is the ratio of the new amplitude to the original amplitude, which is \(4\).
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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
The amplitude of a sine function y = a sin x is the absolute value of the coefficient 'a'. It represents the maximum vertical distance from the midline (usually the x-axis) to the peak of the wave, indicating how tall or stretched the graph is.
Recommended video:
5:05
Amplitude and Reflection of Sine and Cosine
Vertical Stretching of Graphs
Vertical stretching occurs when the output values of a function are multiplied by a factor greater than 1, increasing the distance from the x-axis. For y = a sin x, if |a| > 1, the graph is stretched vertically by a factor of |a| compared to y = sin x.
Recommended video:
6:02Stretches and Shrinks of Functions
Basic Sine Function Graph
The basic sine function y = sin x has an amplitude of 1 and a period of 2π. Understanding its shape and properties is essential to recognize how changes in coefficients affect the graph, such as vertical stretches or compressions.
Recommended video:
5:53Graph of Sine and Cosine Function
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 frequency?
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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Textbook Question
Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = -2 sin 2 πx
Textbook Question
Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = ½ cos π x
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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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