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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 25
Chapter 2, Problem 25

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

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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 \(C\) affects the phase shift.
Find the amplitude by taking the absolute value of the coefficient in front of the sine function: \(\text{Amplitude} = |A| = |-2|\).
Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\) inside the sine function. Here, \(B = 2\).
Determine the phase shift by solving \(Bx + C = 0\) for \(x\), which gives \(x = -\frac{C}{B}\). Substitute \(C = \frac{\pi}{2}\) and \(B = 2\) to find the phase shift.
To graph one period of the function, start at the phase shift on the x-axis, plot key points at intervals of \(\frac{\text{Period}}{4}\), and use the amplitude to mark the maximum and minimum values of the sine wave.

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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. For sine and cosine functions, it is the absolute value of the coefficient before the sine or cosine term. In y = −2 sin(2x + π/2), the amplitude is |−2| = 2.
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Period of a Trigonometric Function

The period is the length of one complete cycle of the function, calculated as 2π divided by the coefficient of x inside the function. For y = −2 sin(2x + π/2), the coefficient is 2, so the period is 2π/2 = π. This means the function repeats every π units.
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Phase Shift of a Trigonometric Function

Phase shift is the horizontal translation of the graph, determined by solving the inside of the function's argument for zero. For y = −2 sin(2x + π/2), set 2x + π/2 = 0, giving x = −π/4. This means the graph shifts left by π/4 units.
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Related Practice
Textbook Question

In Exercises 25–28, use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.

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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 = 1/3 sin 2t

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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 = −8 cos π/2 t

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Textbook Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.

y = 1/2 sin(x + π)

Textbook Question

Use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.


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Textbook Question

In Exercises 1–26, find the exact value of each expression. _ sec⁻¹ (−√2)