1. **State the problem:** We want to graph one period of the function $$y = -3 \cos\left(\frac{1}{2}(x - \frac{\pi}{3})\right) + 1$$ and understand how to move the graph horizontally and vertically. 2. **Identify key parameters:** - Amplitude: $$3$$ (the coefficient before cosine, absolute value) - Period: $$\frac{2\pi}{\frac{1}{2}} = 4\pi$$ - Horizontal phase shift: $$\frac{\pi}{3}$$ units to the right (inside the cosine argument) - Vertical displacement: $$1$$ unit up (added outside cosine) 3. **Explain horizontal shift:** The expression inside cosine is $$\frac{1}{2}(x - \frac{\pi}{3})$$. To find the horizontal shift in terms of $$x$$, set the inside equal to zero: $$\frac{1}{2}(x - \frac{\pi}{3}) = 0 \implies x = \frac{\pi}{3}$$. This means the entire graph shifts $$\frac{\pi}{3}$$ units to the right. 4. **Explain period and scaling:** The period of cosine normally is $$2\pi$$. Because of the factor $$\frac{1}{2}$$ multiplying $$x$$, the period stretches by a factor of $$\frac{1}{\frac{1}{2}} = 2$$, so the new period is $$4\pi$$. 5. **Explain vertical shift:** The $$+1$$ outside the cosine moves the entire graph up by 1 unit. 6. **Graphing steps:** - Start by plotting the baseline at $$y=1$$. - Mark the horizontal axis from $$x=0$$ to $$x=4\pi$$ to cover one full period. - Shift the graph right by $$\frac{\pi}{3}$$ units. - The maximum value is $$1 + 3 = 4$$ and minimum is $$1 - 3 = -2$$. - The cosine wave starts at maximum at $$x=\frac{\pi}{3}$$, goes down to minimum at $$x=\frac{\pi}{3} + 2\pi$$, and back to maximum at $$x=\frac{\pi}{3} + 4\pi$$. 7. **Summary:** - Move the graph horizontally right by $$\frac{\pi}{3}$$. - Stretch the period to $$4\pi$$. - Move the graph vertically up by 1. - Amplitude controls height from baseline. This explains how to move the graph horizontally and vertically and how to plot one period of the function.