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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 23
Chapter 2, Problem 23

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 1/2 sin(x + π/2)

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Identify the general form of the sine function: \(y = A \sin(B(x - C))\), where \(A\) is the amplitude, \(\frac{2\pi}{B}\) is the period, and \(C\) is the phase shift.
Rewrite the given function \(y = \frac{1}{2} \sin(x + \frac{\pi}{2})\) in the form \(y = A \sin(B(x - C))\). Notice that \(x + \frac{\pi}{2}\) can be written as \(x - (-\frac{\pi}{2})\), so \(C = -\frac{\pi}{2}\).
Determine the amplitude \(A\) by looking at the coefficient in front of the sine function. Here, \(A = \frac{1}{2}\), which means the graph oscillates between \(\frac{1}{2}\) and \(-\frac{1}{2}\).
Find the period by identifying \(B\). Since the function is \(\sin(x)\), \(B = 1\), so the period is \(\frac{2\pi}{B} = 2\pi\).
Determine the phase shift \(C\), which is \(-\frac{\pi}{2}\). This means the graph is shifted to the left by \(\frac{\pi}{2}\). Use this information to sketch one full period of the sine wave starting at \(x = -\frac{\pi}{2}\).

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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 a sine or cosine function, representing the height from the midline to the peak. For y = (1/2) sin(x + π/2), the amplitude is 1/2, indicating the wave oscillates between -1/2 and 1/2.
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Period of a Sine Function

The period is the length of one complete cycle of the sine wave, calculated as 2π divided by the coefficient of x inside the function. Since the coefficient of x is 1 here, the period is 2π, meaning the function repeats every 2π units.
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Phase Shift in Trigonometric Functions

Phase shift is the horizontal translation of the graph, determined by solving inside the function for zero. For y = (1/2) sin(x + π/2), the phase shift is -π/2, meaning the graph shifts π/2 units to the left.
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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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Textbook Question

In Exercises 18–24, graph two full periods of the given tangent or cotangent function. y = − 1/2 cot π/2 x

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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

In Exercises 1–26, find the exact value of each expression. _ csc⁻¹ (− 2√3/3)

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Textbook Question

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

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)