Ch. 4 - Graphs of the Circular Functions

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 4 - Graphs of the Circular Functions

Problema 21

Capitolo 5, Problema 21

Match each function in Column I with the appropriate description in Column II.

I
y = -4 sin(3x - 2)

II
A. amplitude = 2, period = π/2, phase shift = ¾
B. amplitude = 3, period = π, phase shift = 2
C. amplitude = 4, period = 2π/3, phase shift = ⅔
D. amplitude = 2, period = 2π/3, phase shift = 4⁄3

Guida verificata passo dopo passo

Identify the general form of the sine function: \(y = A \sin(Bx - C)\), where \(A\) is the amplitude, \(B\) affects the period, and \(C\) affects the phase shift.

Determine the amplitude by taking the absolute value of the coefficient in front of the sine function: \(|A| = |-4| = 4\).

Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Here, \(B = 3\), so the period is \(\frac{2\pi}{3}\).

Find the phase shift using the formula \(\text{Phase shift} = \frac{C}{B}\). Since the function is \(y = -4 \sin(3x - 2)\), rewrite the inside as \(3x - 2 = 3(x - \frac{2}{3})\), so the phase shift is \(\frac{2}{3}\).

Match the calculated values: amplitude = 4, period = \(\frac{2\pi}{3}\), phase shift = \(\frac{2}{3}\) with the options in Column II to find the correct description.

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Amplitude of a Sine Function

Amplitude is the absolute value of the coefficient before the sine function and represents the maximum vertical displacement from the midline. For y = -4 sin(3x - 2), the amplitude is |−4| = 4, indicating the wave oscillates 4 units above and below the center line.

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5:05

Amplitude and Reflection of Sine and Cosine

Period of a Sine Function

The period is the length of one complete cycle of the sine wave and is calculated by dividing 2π by the coefficient of x inside the function. Here, the period is 2π/3 because the coefficient of x is 3, meaning the function repeats every 2π/3 units.

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5:33

Period of Sine and Cosine Functions

Phase Shift of a Sine Function

Phase shift is the horizontal translation of the sine curve and is found by solving the inside of the function's argument for zero: 3x - 2 = 0, so x = 2/3. This means the graph shifts right by 2/3 units, adjusting where the wave starts along the x-axis.

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Phase Shifts

Pratica correlata

Domanda del libro di testo

Identify the circular function that satisfies each description.

period is π; function is decreasing on the interval (0, π)

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Domanda del libro di testo

Match each function in Column I with the appropriate description in Column II.

y = 3 sin(2x - 4)

A. amplitude = 2, period = π/2, phase shift = ¾

B. amplitude = 3, period = π, phase shift = 2

C. amplitude = 4, period = 2π/3, phase shift = ⅔

D. amplitude = 2, period = 2π/3, phase shift = 4⁄3

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Domanda del libro di testo

Each function graphed is of the form y = c + cos x, y = c + sin x, y = cos(x - d), or y = sin(x - d), where d is the least possible positive value. Determine an equation of the graph.

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Domanda del libro di testo

Graph each function over a one-period interval.

y = ½ cot (4x)

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Domanda del libro di testo

Graph each function over a one-period interval.

y = -2 tan (¼ x)

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