Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problem 17

Chapter 2, Problem 17

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

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Identify the general form of the sine function: \(y = a \sin(b(x - c))\), where \(a\) is the amplitude, \(b\) affects the period, and \(c\) is the phase shift.

Compare the given function \(y = \sin(x - \pi)\) to the general form. Here, \(a = 1\), \(b = 1\), and \(c = \pi\).

Calculate the amplitude using the formula \(|a|\). Since \(a = 1\), the amplitude is \(|1| = 1\).

Calculate the period using the formula \(\frac{2\pi}{|b|}\). Since \(b = 1\), the period is \(\frac{2\pi}{1} = 2\pi\).

Determine the phase shift, which is \(c\). The phase shift is \(\pi\) units to the right because the function is \(\sin(x - \pi)\).

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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 = sin(x − π), the amplitude is 1, since the coefficient of sine is 1, indicating the wave oscillates between -1 and 1.

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. For y = sin(x − π), the coefficient is 1, so the period is 2π, meaning the function repeats every 2π units.

Phase Shift of a Trigonometric Function

Phase shift is the horizontal translation of the graph, determined by the value subtracted from x inside the function. In y = sin(x − π), the phase shift is π units to the right, shifting the entire sine curve π units along the x-axis.

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Phase Shifts

Related Practice

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

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

y = sin (x − π/2)

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

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In Exercises 17–24, graph two periods of the given cotangent function. y = 1/2 cot 2x

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In Exercises 7–16, determine the amplitude and period of each function. Then graph one period of the function. y = -sin 2/3 x

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Textbook Question

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