1. **Problem Statement:** We need to sketch the graph of the function $f(x) = 1 + \cos x$. 2. **Formula and Important Rules:** The cosine function $\cos x$ has a range of $[-1,1]$ and a period of $2\pi$. Adding 1 shifts the entire graph of $\cos x$ upward by 1 unit. 3. **Intermediate Work:** - The original cosine function oscillates between $-1$ and $1$. - After shifting up by 1, the new range becomes: $$ [-1 + 1, 1 + 1] = [0, 2] $$ - The period remains $2\pi$ because horizontal shifts or vertical shifts do not affect the period. 4. **Graph Characteristics:** - The maximum value is $2$ at points where $\cos x = 1$, i.e., at $x = 2k\pi$ for integers $k$. - The minimum value is $0$ at points where $\cos x = -1$, i.e., at $x = (2k+1)\pi$. - The midline is $y=1$. 5. **Explanation:** This function is a cosine wave shifted up by 1 unit. It oscillates smoothly between 0 and 2, repeating every $2\pi$ units along the x-axis. 6. **Final Answer:** The graph of $f(x) = 1 + \cos x$ is a cosine wave shifted up by 1, with amplitude 1, period $2\pi$, and range $[0,2]$.