1. **State the problem:** We need to sketch the graph of the function $$y = \cos\left(\frac{x}{2}\right) + 1$$ including two full periods and correct axis markings. 2. **Formula and important rules:** The general cosine function is $$y = A \cos(Bx - C) + D$$ where: - Amplitude = $$|A|$$ - Period = $$\frac{2\pi}{|B|}$$ - Midline = $$y = D$$ - Phase shift = $$\frac{C}{B}$$ For our function, $$A = 1$$, $$B = \frac{1}{2}$$, $$C = 0$$, and $$D = 1$$. 3. **Calculate the period:** $$\text{Period} = \frac{2\pi}{|B|} = \frac{2\pi}{\frac{1}{2}} = 4\pi$$ 4. **Graph characteristics:** - Amplitude = 1 - Midline = $$y = 1$$ - Period = $$4\pi$$ - Starts at maximum when $$x=0$$ because $$\cos(0) = 1$$, so $$y = 1 + 1 = 2$$ - Minimum at $$x = 2\pi$$ because $$\cos\left(\frac{2\pi}{2}\right) = \cos(\pi) = -1$$, so $$y = -1 + 1 = 0$$ 5. **Sketch two full periods:** - One full period spans $$4\pi$$, so two full periods span $$8\pi$$. - Mark x-axis at $$0, 2\pi, 4\pi, 6\pi, 8\pi$$. - Mark y-axis with midline at 1, maximum at 2, minimum at 0. 6. **Summary:** The graph oscillates between 0 and 2, centered at 1, with period $$4\pi$$, starting at maximum at $$x=0$$, minimum at $$x=2\pi$$, maximum at $$x=4\pi$$, and repeats. **Final answer:** The function $$y = \cos\left(\frac{x}{2}\right) + 1$$ has amplitude 1, midline $$y=1$$, period $$4\pi$$, and starts at maximum at $$x=0$$.