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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblem 21
Chapter 2, Problem 21

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 3 sin(2x − π)

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Identify the general form of the sine function: \(y = A \sin(Bx - C)\), where \(A\) is the amplitude, \(B\) affects the period, and \(C\) relates to the phase shift.
Find the amplitude \(A\) by taking the absolute value of the coefficient in front of the sine function. Here, \(A = |3|\).
Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). In this function, \(B = 2\), so substitute to find the period.
Determine the phase shift by solving \(Bx - C = 0\) for \(x\). The phase shift is \(\frac{C}{B}\). Here, \(C = \pi\), so calculate \(\frac{\pi}{2}\).
To graph one period, start at the phase shift on the x-axis, then plot the sine wave over one full period length calculated, using the amplitude to set the maximum and minimum values on the y-axis.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Amplitude of a Trigonometric Function

Amplitude is the maximum absolute value of the function's output, representing the height from the midline to the peak of the wave. For y = a sin(bx + c), the amplitude is |a|. In this case, the amplitude is 3, indicating the wave oscillates 3 units above and below the midline.
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Period of a Sine Function

The period is the length of one complete cycle of the sine wave. It is calculated as (2π) divided by the absolute value of the coefficient b in y = a sin(bx + c). Here, with b = 2, the period is π, meaning the sine function repeats every π units along the x-axis.
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Phase Shift of a Trigonometric Function

Phase shift is the horizontal translation of the graph, determined by solving bx + c = 0 for x. It equals -c/b, indicating how far the graph shifts left or right. For y = 3 sin(2x − π), the phase shift is π/2 units to the right, shifting the wave horizontally.
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Related Practice
Textbook Question

In Exercises 21–28, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, find the following: a. the maximum displacement b. the frequency c. the time required for one cycle. d = 10 cos 2πt

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In Exercises 18–24, graph two full periods of the given tangent or cotangent function. y = −2 tan π/4 x

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In Exercises 17–24, graph two periods of the given cotangent function. y = −3 cot π/2 x

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In Exercises 1–26, find the exact value of each expression. _ cot⁻¹ (−√3)