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

Verified step by step guidance

1. 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.
2. Determine the amplitude by taking the absolute value of the coefficient in front of the sine function: \(|A| = |-4| = 4\).
3. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Here, \(B = 3\), so the period is \(\frac{2\pi}{3}\).
4. 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 \(\frac{2}{3}\).
5. 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.

Key Concepts

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.

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.

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.