Domanda del libro di testo
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -½ cos 3x
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Ch. 4 - Graphs of the Circular Functions
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 5, Problema 6
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = tan 3x
Guida verificata passo dopo passo1
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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Concetti chiave
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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.
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Percorso guidato
6:04
Introduction to Trigonometric Functions
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.
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Percorso guidato
5:43Introduction to Tangent Graph
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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Percorso guidato
6:31Phase Shifts
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