a horizontal compression to produce a period of $\frac{2pi}{3}$ and a horizontal translation of…

Explanation:

Step1: Recall period - change formula

For a trigonometric function (y = A\sin(Bx - C)+D) or (y = A\cos(Bx - C)+D), the period (T=\frac{2\pi}{|B|}). The original period of (y = \sin x) or (y=\cos x) is (T_0 = 2\pi). If the new period (T=\frac{2\pi}{3}), then (\frac{2\pi}{|B|}=\frac{2\pi}{3}), so (|B| = 3>1), which means a horizontal compression. If (T=\frac{3\pi}{2}), then (\frac{2\pi}{|B|}=\frac{3\pi}{2}), and (|B|=\frac{4}{3}<1), which means a horizontal stretch.

Step2: Recall horizontal - translation formula

The horizontal - translation of a function (y = f(x)) to (y = f(x + h)) is (h) units to the left when (h>0) and (h) units to the right when (h < 0). A translation of (\frac{\pi}{4}) units to the left means the function is of the form (y=f(x+\frac{\pi}{4})).

Answer:

a horizontal compression to produce a period of (\frac{2\pi}{3}) and a horizontal translation of (\frac{\pi}{4}) units to the left