Question

e a sine function that has an amplitude of 4, a midline of ( y = 2 ) and a period of ( \frac{2pi}{3} ).

Explanation:

Step1: Recall the general form of a sine function

The general form of a sine function is \(y = A\sin(Bx - C)+D\).

Step2: Determine the amplitude \(A\)

Given the amplitude is \(4\), so \(A = 4\).

Step3: Determine the mid - line \(D\)

Given the mid - line \(y = 2\), so \(D=2\).

Step4: Determine the period formula and find \(B\)

The period formula for \(y = A\sin(Bx - C)+D\) is \(T=\frac{2\pi}{|B|}\). Given \(T = \frac{2\pi}{3}\), then \(\frac{2\pi}{|B|}=\frac{2\pi}{3}\), so \(|B| = 3\). We can take \(B = 3\) (assuming no phase shift \(C = 0\) for simplicity).

Answer:

\(y=4\sin(3x)+2\)