Domanda del libro di testo
Match each function with its graph in choices A–I. (One choice will not be used.)
y = sin (x - π/4)
A. <IMAGE> B. <IMAGE> C. <IMAGE>
D. <IMAGE> E. <IMAGE> F. <IMAGE>
G. <IMAGE> H. <IMAGE> I. <IMAGE>
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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 9
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 1 + 2 sin ¼ x
Guida verificata passo dopo passo1
Identify the general form of the sine function: \(y = A \sin(B(x - C)) + D\), where \(A\) is the amplitude, \(\frac{2\pi}{B}\) is the period, \(C\) is the phase shift, and \(D\) is the vertical translation.
Compare the given function \(y = 1 + 2 \sin \frac{1}{4} x\) to the general form. Rewrite it as \(y = 2 \sin \left( \frac{1}{4} x \right) + 1\) to clearly see the components.
Determine the amplitude \(A\) by looking at the coefficient in front of the sine function. Here, \(A = 2\).
Calculate the period using the formula \(\text{Period} = \frac{2\pi}{B}\), where \(B = \frac{1}{4}\). So, the period is \(\frac{2\pi}{\frac{1}{4}}\).
Identify the vertical translation \(D\) as the constant added outside the sine function, which is \(1\). Since there is no subtraction or addition inside the sine argument, the phase shift \(C\) is \(0\).
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Concetti chiave
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Amplitude of a Trigonometric Function
Amplitude measures the maximum distance a sine or cosine function's graph reaches from its midline. It is the absolute value of the coefficient before the sine or cosine term. For example, in y = 2 sin(¼x), the amplitude is 2, indicating the wave oscillates 2 units above and below its central axis.
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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 sine or cosine function. It is calculated by dividing 2π by the absolute value of the coefficient of x inside the function. For y = 2 sin(¼x), the period is 2π ÷ ¼ = 8π, meaning the function repeats every 8π units along the x-axis.
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Percorso guidato
5:33Period of Sine and Cosine Functions
Vertical Translation and Phase Shift
Vertical translation shifts the graph up or down and is determined by the constant added outside the sine or cosine function, here +1. Phase shift moves the graph left or right and depends on horizontal shifts inside the function's argument. In y = 1 + 2 sin(¼x), the vertical translation is +1, and there is no phase shift since there is no horizontal addition or subtraction inside the sine.
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Percorso guidato
6:31Phase Shifts
Pratica correlata
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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Domanda del libro di testo
Match each function with its graph in choices A–F.
y = tan (x - π )
A. <IMAGE> B. <IMAGE> C. <IMAGE>
D. <IMAGE> E. <IMAGE> F. <IMAGE>
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Domanda del libro di testo
Fill in the blank(s) to correctly complete each sentence.
The graph of y = -2 + 3 cos (x - π/6) 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)
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Domanda del libro di testo
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 3 cos (x + π/2)
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Domanda del libro di testo
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 3 - ¼ cos ⅔ x
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