1. **State the problem:** We are given a sinusoidal function of the form $y = a \sin(bx)$ or $y = a \cos(bx)$ and a graph that oscillates between -4 and 4 on the y-axis with zeros at multiples of approximately $-\frac{4\pi}{3}$, $-\frac{2\pi}{3}$, $\frac{2\pi}{3}$, $\frac{4\pi}{3}$. 2. **Identify amplitude $a$:** The amplitude is the maximum absolute value of $y$, which is 4. So, $a = 4$. 3. **Identify zeros and period:** The zeros occur at $x = -\frac{4\pi}{3}, -\frac{2\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}$. The distance between consecutive zeros is $\frac{2\pi}{3}$. 4. **Relate zeros to period for sine function:** For $y = a \sin(bx)$, zeros occur at $x = \frac{k\pi}{b}$ for integer $k$. Given zeros at $x = \pm \frac{2\pi}{3}, \pm \frac{4\pi}{3}$, the spacing between zeros is $\frac{\pi}{b} = \frac{2\pi}{3}$. 5. **Solve for $b$:** $$\frac{\pi}{b} = \frac{2\pi}{3} \implies b = \frac{3}{2}$$ 6. **Write the function:** Since the graph resembles a sine wave centered at zero with amplitude 4 and frequency $b = \frac{3}{2}$, the function is: $$y = 4 \sin\left(\frac{3}{2} x\right)$$ **Final answer:** $$y = 4 \sin\left(\frac{3}{2} x\right)$$