1. **State the problem:** We are given the function $$y = \sqrt{3} \sin x - 3 \cos x + 4$$ and want to understand its transformation from the basic sine function $$y = \sin x$$. 2. **Recall the formula for combining sine and cosine:** Any expression of the form $$a \sin x + b \cos x$$ can be rewritten as $$R \sin(x + \phi)$$ where $$R = \sqrt{a^2 + b^2}$$ and $$\phi = \arctan\left(\frac{b}{a}\right)$$. 3. **Calculate amplitude $$R$$:** $$R = \sqrt{(\sqrt{3})^2 + (-3)^2} = \sqrt{3 + 9} = \sqrt{12} = 2\sqrt{3}$$. 4. **Calculate phase shift $$\phi$$:** $$\phi = \arctan\left(\frac{-3}{\sqrt{3}}\right) = \arctan(-\sqrt{3}) = -\frac{\pi}{3}$$. 5. **Rewrite the function:** $$y = 2\sqrt{3} \sin\left(x - \frac{\pi}{3}\right) + 4$$. 6. **Interpretation:** - The amplitude is stretched from 1 to $$2\sqrt{3}$$. - The graph is shifted to the right by $$\frac{\pi}{3}$$. - The entire graph is translated upward by 4 units. This matches the description of vertical stretch, phase shift, and vertical translation. **Final answer:** $$y = 2\sqrt{3} \sin\left(x - \frac{\pi}{3}\right) + 4$$