1. **State the problem:** We need to analyze and sketch the graph of the function $$f(x) = 1 + \cos x$$. 2. **Recall the cosine function properties:** The cosine function $$\cos x$$ has a range of $$[-1,1]$$ and a period of $$2\pi$$. It oscillates between -1 and 1, with maximum at $$x=2k\pi$$ and minimum at $$x=\pi + 2k\pi$$ for integers $$k$$. 3. **Apply the transformation:** The function $$f(x) = 1 + \cos x$$ shifts the cosine graph vertically upward by 1 unit. 4. **Determine the new range:** Since $$\cos x$$ ranges from -1 to 1, adding 1 shifts the range to $$[-1+1, 1+1] = [0, 2]$$. 5. **Identify key points:** - At $$x=0$$, $$f(0) = 1 + \cos 0 = 1 + 1 = 2$$ (maximum). - At $$x=\pi$$, $$f(\pi) = 1 + \cos \pi = 1 - 1 = 0$$ (minimum). - At $$x=\frac{\pi}{2}$$, $$f\left(\frac{\pi}{2}\right) = 1 + \cos \frac{\pi}{2} = 1 + 0 = 1$$ (midline). 6. **Periodicity:** The function repeats every $$2\pi$$. 7. **Sketching the graph:** - Draw the horizontal axis $$x$$ and vertical axis $$y$$. - Mark the key points: maxima at $$x=0, 2\pi, 4\pi, ...$$ with value 2. - Minima at $$x=\pi, 3\pi, 5\pi, ...$$ with value 0. - The midline is at $$y=1$$. - The graph oscillates smoothly between 0 and 2, following the cosine shape shifted up by 1. **Final answer:** The graph of $$f(x) = 1 + \cos x$$ is a cosine wave shifted up by 1 unit, with range $$[0,2]$$ and period $$2\pi$$.