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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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 4.29
Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = 2 sin ¼ x
Verified step by step guidance1
Identify the standard form of the sine function: \( y = a \sin(bx + c) + d \). In this case, \( a = 2 \), \( b = \frac{1}{4} \), \( c = 0 \), and \( d = 0 \).
Determine the amplitude of the function. The amplitude is the absolute value of \( a \), which is \( |2| = 2 \).
Calculate the period of the function. The period of a sine function is given by \( \frac{2\pi}{b} \). Substitute \( b = \frac{1}{4} \) to find the period: \( \frac{2\pi}{\frac{1}{4}} = 8\pi \).
Graph the function over a two-period interval. Since one period is \( 8\pi \), a two-period interval is \( 16\pi \). Plot the function from \( x = 0 \) to \( x = 16\pi \).
Sketch the sine wave, noting that it oscillates between \( -2 \) and \( 2 \) (the amplitude) and completes one full cycle every \( 8\pi \) units along the x-axis.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Period of a Trigonometric Function
The period of a trigonometric function is the length of one complete cycle of the wave. For the sine function, the standard period is 2π. However, when the function is modified, such as in y = 2 sin(¼ x), the period can be calculated using the formula Period = 2π / |b|, where b is the coefficient of x. In this case, the period is 2π / (1/4) = 8π.
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Period of Sine and Cosine Functions
Amplitude of a Trigonometric Function
The amplitude of a trigonometric function refers to the maximum distance the function reaches from its midline. For the sine function, the amplitude is determined by the coefficient in front of the sine term. In the function y = 2 sin(¼ x), the amplitude is 2, meaning the graph will oscillate between 2 and -2.
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Graphing Trigonometric Functions
Graphing trigonometric functions involves plotting the values of the function over a specified interval. For y = 2 sin(¼ x), one must consider the calculated period and amplitude to accurately represent the wave. The graph will show oscillations between the maximum and minimum values defined by the amplitude, repeating every 8π units along the x-axis.
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Related Practice
Textbook Question
Graph each function over the interval [-2π, 2π]. Give the amplitude. See Example 1.
y = 2 cos x
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
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 2 sin 2x
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Textbook Question
Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = π sin πx
Textbook Question
Determine an equation for each graph.