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


I
y = 3 sin(2x - 4)


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. From the given function \(y = 3 \sin(2x - 4)\), note that the amplitude \(A\) is the coefficient before the sine, which is 3.
3. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{B}\). Here, \(B = 2\), so the period is \(\frac{2\pi}{2} = \pi\).
4. Find the phase shift using the formula \(\text{Phase shift} = \frac{C}{B}\). Since the function is \(\sin(2x - 4)\), rewrite the inside as \$2(x - 2)$, so the phase shift is \(\frac{4}{2} = 2\).
5. Match the values amplitude = 3, period = \(\pi\), and phase shift = 2 with the correct description in Column II, which corresponds to option B.

Key Concepts

Amplitude of a Sine Function

Amplitude is the maximum value or height of the sine wave from its midline. For a function y = a sin(bx + c), the amplitude is the absolute value of 'a'. It determines how far the graph stretches vertically from the center line.

Period of a Sine Function

The period is the length of one complete cycle of the sine wave. It is calculated as (2π) divided by the absolute value of the coefficient 'b' in y = a sin(bx + c). The period tells how frequently the wave repeats over the x-axis.

Phase Shift of a Sine Function

Phase shift is the horizontal translation of the sine graph, determined by solving (bx + c) = 0 for x. It equals -c/b and indicates how far the graph shifts left or right from the origin. This shift affects where the wave starts on the x-axis.