Question

match each function in column i with the appropriate description in column ii.
type the letter of the correct answer choice.
i
$y = 2\sin(3x - 4)$ is described by choice
$y = 3\sin(2x - 4)$ is described by choice
$y = - 4\sin(2x - 3)$ is described by choice
$y = - 3\sin(4x - 2)$ is described by choice
ii
a. amplitude $= 3$, period $=\pi$, phase shift $= 2$
b. amplitude $= 2$, period $=\frac{2\pi}{3}$, phase shift $=\frac{4}{3}$
c. amplitude $= 3$, period $=\frac{\pi}{2}$, phase shift $=\frac{1}{2}$
d. amplitude $= 4$, period $=\pi$, phase shift $=\frac{3}{2}$

Explanation:

Step1: Recall the formula for sinusoidal functions

For a function \(y = A\sin(Bx - C)\), the amplitude is \(|A|\), the period is \(\frac{2\pi}{B}\), and the phase shift is \(\frac{C}{B}\).

Step2: Analyze \(y = 2\sin(3x - 4)\)

• Amplitude: \(|2| = 2\)
• Period: \(\frac{2\pi}{3}\)
• Phase shift: \(\frac{4}{3}\)

This matches option B.

Step3: Analyze \(y = 3\sin(2x - 4)\)

• Amplitude: \(|3| = 3\)
• Period: \(\frac{2\pi}{2}=\pi\)
• Phase shift: \(\frac{4}{2} = 2\)

But there is no such option. Wait, re - check formula. Wait, no, for \(y = A\sin(Bx - C)\), phase shift is \(\frac{C}{B}\). For \(y = 3\sin(2x - 4)\), \(A = 3\), \(B=2\), \(C = 4\). Amplitude \(|A|=3\), period \(\frac{2\pi}{B}=\pi\), phase shift \(\frac{C}{B}=2\). But looking at options, for \(y = 3\sin(2x - 4)\) there is no option. Wait, maybe a typo. Wait, no, let's check other functions.

Step4: Analyze \(y=-4\sin(2x - 3)\)

• Amplitude: \(|-4| = 4\)
• Period: \(\frac{2\pi}{2}=\pi\)
• Phase shift: \(\frac{3}{2}\)

This matches option D.

Step5: Analyze \(y=-3\sin(4x - 2)\)

• Amplitude: \(|-3| = 3\)
• Period: \(\frac{2\pi}{4}=\frac{\pi}{2}\)
• Phase shift: \(\frac{2}{4}=\frac{1}{2}\)

This matches option C.

Answer:

For \(y = 2\sin(3x - 4)\): B
For \(y = 3\sin(2x - 4)\): (No correct option in given. But if we assume a mis - labeling, maybe there was an error in problem creation. But based on standard formula)
For \(y=-4\sin(2x - 3)\): D
For \(y=-3\sin(4x - 2)\): C