1. **State the problem:** We need to find an equation of the form $y = a \sin(bx)$ or $y = a \cos(bx)$ that matches the given sinusoidal graph. 2. **Identify amplitude $a$:** The graph oscillates between approximately $4$ and $-4$, so the amplitude is half the distance between max and min: $$a = \frac{4 - (-4)}{2} = \frac{8}{2} = 4$$ 3. **Identify period $T$ and frequency $b$:** The graph completes one full cycle from $x=0$ to $x=2\pi$, so the period is: $$T = 2\pi$$ The frequency $b$ relates to the period by: $$b = \frac{2\pi}{T} = \frac{2\pi}{2\pi} = 1$$ 4. **Determine phase shift and function type:** The graph starts near $y=3.5$ at $x=0$ and decreases, which resembles a cosine wave shifted slightly or a sine wave with phase shift. Since cosine at $x=0$ is $a$, and here $y \approx 3.5$ close to $4$, cosine fits well. 5. **Write the equation:** Using cosine with amplitude $4$ and frequency $1$: $$y = 4 \cos(x)$$ 6. **Verify:** At $x=0$, $y=4\cos(0)=4$, close to the observed $3.5$ (small deviation likely due to graph scale). At $x=\pi$, $y=4\cos(\pi)=-4$, matching the trough. At $x=2\pi$, $y=4\cos(2\pi)=4$, matching the peak. **Final answer:** $$y = 4 \cos(x)$$