Describe the transformation of the graph when the equation of the cosine function changes from \( y = \cos(x) \) to \( y = \cos(x - \frac{\pi}{3}) + 2 \). What effects does this have on the graph?

When transforming the equation of the cosine function from \( y = \cos(x) \) to \( y = \cos(x - \frac{\pi}{3}) + 2 \), you first notice a horizontal shift. The term \( x - \frac{\pi}{3} \) indicates the graph shifts to the right by \( \frac{\pi}{3} \), meaning every point on the original cosine curve moves \( \frac{\pi}{3} \) units to the right. Next, the addition of \( +2 \) at the end lifts the entire graph up by 2 units. This results in a transformation where not only is the wave shifted right, but it also oscillates around the line \( y = 2 \) instead of around the x-axis, giving you a new midline. So, the peaks of the wave reach up to 3, while the troughs dip to 1, reflecting this new vertical shift!