1. **Stating the problem:** We are given the function $y = 2 \cos x$ and want to understand its behavior and graph. 2. **Formula and explanation:** The cosine function $\cos x$ oscillates between $-1$ and $1$ with period $2\pi$. Multiplying by 2 scales the amplitude to oscillate between $-2$ and $2$. 3. **Key properties:** - Amplitude: $2$ - Period: $2\pi$ - The function starts at $y=2$ when $x=0$ because $\cos 0 = 1$ - It crosses the x-axis when $\cos x = 0$, i.e., at $x = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots$ 4. **Intermediate work:** - At $x=0$, $y=2 \cos 0 = 2 \times 1 = 2$ - At $x=\frac{\pi}{3}$, $y=2 \cos \frac{\pi}{3} = 2 \times \frac{1}{2} = 1$ - At $x=\frac{2\pi}{3}$, $y=2 \cos \frac{2\pi}{3} = 2 \times (-\frac{1}{2}) = -1$ - At $x=\pi$, $y=2 \cos \pi = 2 \times (-1) = -2$ 5. **Explanation:** The graph oscillates smoothly between $2$ and $-2$ with zeros at $x=\frac{\pi}{2}, \frac{3\pi}{2}, \ldots$. The x-axis labeling from $0$ to $\frac{10\pi}{3}$ covers multiple periods. **Final answer:** The function $y=2\cos x$ is a cosine wave with amplitude 2 and period $2\pi$, oscillating between $-2$ and $2$ as described.