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

if the function ( y = sin x ) is transformed to ( y = 3sinleft(\frac{2}{3}x\right) ), 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 = 3sinleft(\frac{2}{3}x\right) ), 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: Find the amplitude

For the function (y = A\sin(Bx)), the amplitude is (|A|). For (y=\sin x), (A = 1), amplitude (= 1). For (y = 3\sin(\frac{2}{3}x)), (A = 3), amplitude (= 3). Since (3>1), the amplitude increases.

Step2: Find the period

The period of (y=\sin x) is (T_1=2\pi) (because for (y = \sin x), (B = 1) and period (T=\frac{2\pi}{B})). For (y = 3\sin(\frac{2}{3}x)), (B=\frac{2}{3}), and the period (T_2=\frac{2\pi}{B}=\frac{2\pi}{\frac{2}{3}}=3\pi). Since (3\pi>2\pi), the period increases.

Answer:

The amplitude increases, and the period increases.