Ch. 4 - Graphs of the Circular Functions

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 4 - Graphs of the Circular Functions

Problem 6

Chapter 5, Problem 6

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = tan 3x

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Identify the general form of the tangent function: \(y = a \tan(bx - c) + d\), where \(a\) is the amplitude, \(b\) affects the period, \(c\) is the phase shift, and \(d\) is the vertical translation.

Note that the tangent function does not have an amplitude because its values range from \(-\infty\) to \(+\infty\), so amplitude is not defined for \(y = \tan 3x\).

Determine the period using the formula for tangent: \(\text{Period} = \frac{\pi}{|b|}\). Here, \(b = 3\), so the period is \(\frac{\pi}{3}\).

Check for vertical translation \(d\). Since there is no constant added or subtracted outside the tangent function, the vertical translation is \(0\).

Find the phase shift using \(\frac{c}{b}\). Since the function is \(y = \tan 3x\) with no subtraction inside the argument, \(c = 0\), so the phase shift is \(0\).

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Key Concepts

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

Amplitude of Trigonometric Functions

Amplitude measures the maximum vertical distance from the midline to the peak of a wave. For sine and cosine functions, amplitude is the absolute value of the coefficient before the function. However, tangent functions do not have an amplitude because their values increase without bound.

Period of the Tangent Function

The period of a function is the length of one complete cycle on the x-axis. The basic tangent function, y = tan x, has a period of π. When the function is y = tan(bx), the period changes to π divided by the absolute value of b, compressing or stretching the graph horizontally.

Phase Shift and Vertical Translation

Phase shift refers to the horizontal shift of the graph, determined by any added or subtracted value inside the function's argument. Vertical translation shifts the graph up or down, determined by any constant added or subtracted outside the function. In y = tan 3x, there is no phase shift or vertical translation since no such terms are present.

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Phase Shifts

Related Practice

Textbook Question

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.

y = -½ cos 3x

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Textbook Question

Graph each function over the interval [-2π, 2π]. Give the amplitude. See Example 1.

y = 2 cos x

Textbook Question

Fill in the blank(s) to correctly complete each sentence.

The graph of y = 6 + 3 sin x is obtained by shifting the graph of y = 3 sin x ________ unit(s) __________ (up/down).

Textbook Question

Fill in the blank(s) to correctly complete each sentence.

The graph of y = 3 + 5 cos (x + π/5) is obtained by shifting the graph of y = cos x horizontally ________ unit(s) to the __________, (right/left) stretching it vertically by a factor of ________, and then shifting it vertically ________ unit(s) __________. (up/down)

Textbook Question

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.

y = 2 sin 2x

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Textbook Question

Fill in the blank(s) to correctly complete each sentence.

The graph of y = -5 + 2 cos x is obtained by shifting the graph of y = 2 cos x ________ unit(s) __________ (up/down).

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For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.y = tan 3x

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Key Concepts

Amplitude of Trigonometric Functions

Period of the Tangent Function

Phase Shift and Vertical Translation

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = tan 3x