Textbook Question
Determine an equation for each graph.
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.
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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.
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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.
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6:31Phase Shifts
Related Practice
Textbook Question
Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = 2 sin ¼ x
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.
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.
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.