Cosine 2X 474Cd5

1. **State the problem:** We need to sketch the graph of the function $y= \cos(2x)$. 2. **Recall the formula and properties:** The general cosine function is $y= \cos(bx)$ where $b$ affects the period of the function. The period $T$ is given by $$T = \frac{2\pi}{|b|}.$$ 3. **Calculate the period:** Here, $b=2$, so the period is $$T = \frac{2\pi}{2} = \pi.$$ This means the graph completes one full cycle every $\pi$ units along the x-axis, which is half the usual period of $2\pi$ for $\cos x$. 4. **Key points for one period:** For $y=\cos(2x)$, the key points in one period $[0, \pi]$ are: - At $x=0$, $y=\cos(0)=1$. - At $x=\frac{\pi}{4}$, $y=\cos(2 \cdot \frac{\pi}{4})=\cos(\frac{\pi}{2})=0$. - At $x=\frac{\pi}{2}$, $y=\cos(\pi)=-1$. - At $x=\frac{3\pi}{4}$, $y=\cos(\frac{3\pi}{2})=0$. - At $x=\pi$, $y=\cos(2\pi)=1$. 5. **Shape and amplitude:** The amplitude remains $1$ (max 1, min -1). The graph oscillates faster than $y=\cos x$ because of the factor 2 inside the cosine. 6. **Sketching:** Plot these points and connect smoothly with the characteristic cosine wave shape repeating every $\pi$. **Final answer:** The graph of $y=\cos(2x)$ is a cosine wave with period $\pi$ and amplitude $1$, oscillating twice as fast as $y=\cos x$.