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 4.42
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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Verified step by step guidance1
Identify the type of trigonometric function: Determine whether the graph resembles a sine or cosine function based on its shape and starting point.
Determine the amplitude 'a': Measure the vertical distance from the midline of the graph to a peak or trough. This value is the amplitude.
Determine the period of the function: Measure the horizontal distance required for the graph to complete one full cycle.
Calculate the value of 'b': Use the formula for the period of a sine or cosine function, which is \( \frac{2\pi}{b} \), and solve for 'b'.
Write the equation: Substitute the values of 'a' and 'b' into the form \( y = a \cos(bx) \) or \( y = a \sin(bx) \), depending on the function type identified in step 1.
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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 height of a wave from its central axis. In the equations y = a cos(bx) or y = a sin(bx), the value 'a' represents the amplitude. It determines how far the graph stretches vertically from the midline, affecting the overall height of the peaks and depth of the troughs.
Recommended video:
5:05
Amplitude and Reflection of Sine and Cosine
Period
The period of a trigonometric function is the distance along the x-axis required for the function to complete one full cycle. In the equations y = a cos(bx) or y = a sin(bx), the period is calculated as 2π/b. This concept is crucial for understanding how frequently the wave oscillates and is essential for matching the graph to the correct function.
Recommended video:
5:33Period of Sine and Cosine Functions
Phase Shift
Phase shift refers to the horizontal displacement of a trigonometric graph. It occurs when the function is adjusted by adding or subtracting a constant inside the argument of the sine or cosine function. Understanding phase shift is important for accurately positioning the graph along the x-axis, which can be necessary to match the given graph in the problem.
Recommended video:
6:31Phase Shifts
Related Practice
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
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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 = -3 sin x is obtained by stretching the graph of y = sin x by a factor of ________ and reflecting across the ________-axis.
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Textbook Question
Decide whether each statement is true or false. If false, explain why.
The graph of y = sec x in Figure 37 suggests that sec(-x) = sec x for all x in the domain of sec x.
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