if the function $y = \\sin x$ is transformed to $y = 3\\sin(\\frac{2}{3}x)$, how do the amplitude and period…

if the function $y = \\sin x$ is transformed to $y = 3\\sin(\\frac{2}{3}x)$, how do the amplitude and period change?\nthe amplitude increases, and the period decreases.\nthe amplitude increases, and the period increases.\nthe amplitude decreases, and the period decreases.\nthe amplitude decreases, and the period increases.

if the function $y = \\sin x$ is transformed to $y = 3\\sin(\\frac{2}{3}x)$, how do the amplitude and period change?\nthe amplitude increases, and the period decreases.\nthe amplitude increases, and the period increases.\nthe amplitude decreases, and the period decreases.\nthe amplitude decreases, and the period increases.

Answer

Explanation:

Step1: Recall amplitude formula

For $y = A\sin(Bx)$, amplitude is $|A|$. For $y=\sin x$, $A = 1$, amplitude is 1. For $y = 3\sin(\frac{2}{3}x)$, $A = 3$, and $3>1$, so amplitude increases.

Step2: Recall period formula

The period of $y=\sin x$ is $T_1 = 2\pi$. For $y = A\sin(Bx)$, period $T=\frac{2\pi}{|B|}$. For $y = 3\sin(\frac{2}{3}x)$, $B=\frac{2}{3}$, and $T_2=\frac{2\pi}{\frac{2}{3}}=3\pi$. Since $3\pi>2\pi$, period increases.

Answer:

The amplitude increases, and the period increases.