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 45

Chapter 2, Problem 45

In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 3 cos(2x − π)

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Identify the general form of the cosine function: \(y = A \cos(Bx - C)\), 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 cosine: \(A = |3|\).

Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\). Here, \(B = 2\), so substitute to get the period.

Determine the phase shift by solving \(Bx - C = 0\) for \(x\), which gives \(x = \frac{C}{B}\). Here, \(C = \pi\), so calculate the phase shift as \(\frac{\pi}{2}\).

Interpret the phase shift direction: since the function is \(\cos(2x - \pi)\), the phase shift is to the right by \(\frac{\pi}{2}\). Then, use these values to sketch one period of the cosine function starting from the phase shift.

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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 cos(bx − c), the amplitude is |a|. In this case, the amplitude is 3, indicating the wave oscillates 3 units above and below the midline.

Period of a Trigonometric Function

The period is the length of one complete cycle of the function along the x-axis. For y = cos(bx − c), the period is calculated as (2π) / |b|. Here, with b = 2, the period is π, meaning the cosine wave repeats every π units.

Phase Shift of a Trigonometric Function

Phase shift is the horizontal translation of the graph, determined by solving bx − c = 0 for x. It is given by c / b. For y = 3 cos(2x − π), the phase shift is (π) / 2 units to the right, indicating the graph is shifted right by π/2 from the standard cosine curve.

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Phase Shifts

Related Practice

Textbook Question

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan (tan⁻¹ 125)

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Textbook Question

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

y = cos(x + π/2)

Textbook Question

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin(cos⁻¹ 3/5)

Textbook Question

In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 1/2 cos (3x + π/2)

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Textbook Question

In Exercises 45–52, graph two periods of each function. y = sec(2x + π/2) − 1

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Textbook Question

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

y = 4 cos(2x − π)

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