Domanda del libro di testo
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 2 sin 5x
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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 7
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -½ cos 3x
Guida verificata passo dopo passo1
Identify the general form of the cosine function: \(y = A \cos(Bx - C) + D\), where \(A\) is the amplitude, \(B\) affects the period, \(C\) is the phase shift, and \(D\) is the vertical translation.
Find the amplitude by taking the absolute value of the coefficient in front of the cosine function: \(|A| = |-\frac{1}{2}|\).
Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\) inside the cosine function.
Determine the vertical translation \(D\) by identifying any constant added or subtracted outside the cosine function. In this case, there is none, so \(D = 0\).
Find the phase shift using the formula \(\text{Phase shift} = \frac{C}{B}\), where \(C\) is the horizontal shift inside the cosine argument. Since there is no \(C\) term, the phase shift is zero.
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Concetti chiave
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Amplitude of a Trigonometric Function
Amplitude measures the maximum distance a trigonometric function's graph reaches from its midline. For functions like y = a cos bx, the amplitude is the absolute value of 'a'. It determines the height of the peaks and depths of the troughs in the wave.
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Percorso guidato
6:04
Introduction to Trigonometric Functions
Period of a Trigonometric Function
The period is the length of one complete cycle of the function. For y = cos(bx), the period is calculated as (2π) divided by the absolute value of 'b'. It indicates how frequently the wave repeats over the x-axis.
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Percorso guidato
5:33Period of Sine and Cosine Functions
Phase Shift and Vertical Translation
Phase shift refers to the horizontal shift of the graph, determined by any horizontal addition or subtraction inside the function's argument. Vertical translation is the upward or downward shift of the graph, indicated by any constant added or subtracted outside the function. In y = -½ cos 3x, there is no phase shift or vertical translation since no horizontal or vertical constants are added.
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Percorso guidato
6:31Phase Shifts
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Fill in the blank(s) to correctly complete each sentence.
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For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = tan 3x
733
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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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