Question

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

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.