Textbook Question
Determine an equation for each graph.
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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 4
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.
Verified step by step guidance1
Identify the general form of the sine function transformation: \(y = A \sin x\), where \(A\) represents the amplitude, which affects the vertical stretch or compression of the graph.
Compare the given function \(y = -3 \sin x\) with the basic sine function \(y = \sin x\). Notice that the amplitude changes from 1 to 3, indicating a vertical stretch by a factor of 3.
Observe the negative sign in front of the amplitude, which means the graph is reflected across the x-axis.
Summarize the transformation: the graph is stretched vertically by a factor of 3 and reflected across the x-axis.
Fill in the blanks accordingly: The graph of \(y = -3 \sin x\) is obtained by stretching the graph of \(y = \sin x\) by a factor of 3 and reflecting across the x-axis.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Amplitude and Vertical Stretching
The amplitude of a sine function is the absolute value of the coefficient before sin x, which determines the vertical stretch or compression. For y = -3 sin x, the amplitude is 3, meaning the graph is stretched vertically by a factor of 3 compared to y = sin x.
Recommended video:
6:02
Stretches and Shrinks of Functions
Reflection Across the x-axis
A negative coefficient before the sine function indicates a reflection of the graph across the x-axis. This flips the graph upside down, changing the direction of the peaks and troughs while preserving the shape.
Recommended video:
5:00Reflections of Functions
Basic Sine Function Graph
The graph of y = sin x is a periodic wave oscillating between -1 and 1 with a period of 2π. Understanding this baseline graph is essential to recognize how transformations like stretching and reflection affect its shape.
Recommended video:
5:53Graph of Sine and Cosine Function
Related Practice
Textbook Question
Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph. See Example 6.
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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
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 2 sin 2x
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Textbook Question
Determine an equation for each graph.
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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).
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