Answer
$y=-4\ cos(3x)$
Work Step by Step
Step 1. As the curve is even and passes $(0,-4)$, we can write a general form $y=-A\ cos(kx)$
Step 2. We can identify the amplitude $A=4$.
Step 3. We can identify the period $p=\frac{2\pi}{3}$, thus $\frac{2\pi}{k}=\frac{2\pi}{3}$ and $k=3$
Step 4. The equation is $y=-4\ cos(3x)$