The curve \(y = \sin x\) is transformed to the curve \(y = 4 \sin\left(\frac{1}{2}x - 30^\circ\right)\).
Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations are applied.
Solution
1. Translate the graph by \(\begin{pmatrix} 30^\circ \\ 0 \end{pmatrix}\) to account for the phase shift of \(-30^\circ\).
2. Stretch the graph by a factor of 2 in the x-direction to account for the coefficient \(\frac{1}{2}\) in front of \(x\).
3. Stretch the graph by a factor of 4 in the y-direction to account for the amplitude change from 1 to 4.
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