Textbook Question
Graph each function over a one-period interval.
y = 3 sec [(1/4)x]
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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 4.15
Graph each function over the interval [-2π, 2π]. Give the amplitude. See Example 1.
y = ⅔ sin x
Verified step by step guidance1
Identify the function type: The given function is y = \(\frac{2}{3}\) \(\sin\) x, which is a sine function.
Determine the amplitude: The amplitude of a sine function y = a \(\sin\) x is the absolute value of the coefficient a. Here, a = \(\frac{2}{3}\), so the amplitude is \(\left\)|\(\frac{2}{3}\)\(\right\)|.
Set the interval for graphing: The problem specifies the interval [-2\(\pi\), 2\(\pi\)]. This means you will graph the function from x = -2\(\pi\) to x = 2\(\pi\).
Plot key points: For the sine function, key points occur at x = 0, \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and 2\(\pi\), and their corresponding negative values. Calculate y for these x-values using y = \(\frac{2}{3}\) \(\sin\) x.
Sketch the graph: Use the key points and the amplitude to sketch the sine wave, ensuring it oscillates between -\(\frac{2}{3}\) and \(\frac{2}{3}\) over the interval [-2\(\pi\), 2\(\pi\)].
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Amplitude
Amplitude refers to the maximum distance a wave reaches from its central position, which is crucial in understanding the height of trigonometric functions like sine and cosine. For the function y = ⅔ sin x, the amplitude is ⅔, indicating that the graph will oscillate between -⅔ and ⅔. This concept helps in visualizing the vertical stretch or compression of the sine wave.
Recommended video:
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Amplitude and Reflection of Sine and Cosine
Graphing Trigonometric Functions
Graphing trigonometric functions involves plotting the values of the function over a specified interval, in this case, [-2π, 2π]. Understanding the periodic nature of sine functions, which repeat every 2π, is essential for accurately representing the function's behavior across the given interval. This includes identifying key points such as intercepts, maximums, and minimums.
Recommended video:
6:04Introduction to Trigonometric Functions
Period of the Sine Function
The period of a sine function is the length of one complete cycle of the wave. For the standard sine function, the period is 2π, meaning it repeats every 2π units along the x-axis. In the function y = ⅔ sin x, the period remains 2π, which is important for determining how many cycles will fit within the interval [-2π, 2π] and for accurately sketching the graph.
Recommended video:
5:33Period of Sine and Cosine Functions
Related Practice
Textbook Question
Graph each function over a one-period interval.
y = sec (x + π/4)
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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
Determine an equation for each graph.
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Textbook Question
Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = sin 3x
Textbook Question
Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = sin ⅔ x
4
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