The original stationary point is at \((p, q)\).

The transformation applied to the curve is \(y = -3f\left(\frac{1}{4}(x + 8)\right)\).

This transformation involves:

• A horizontal translation of \(-8\) units (due to \(x + 8\)).
• A horizontal scaling by a factor of \(4\) (due to \(\frac{1}{4}x\)).
• A vertical scaling by a factor of \(-3\) (due to \(-3f(x)\)).

For the x-coordinate:

Start with the horizontal translation: \(x' = x + 8\), so \(x = x' - 8\).

Apply the horizontal scaling: \(x'' = 4x\), so \(x'' = 4(x' - 8) = 4p - 8\).

For the y-coordinate:

Apply the vertical scaling: \(y'' = -3y\), so \(y'' = -3q\).

Thus, the coordinates of the stationary point for the transformed curve are \((4p - 8, -3q)\).

To transform the graph of \(y = \sin x\) to \(y = f(x) = 3 + 2 \sin \frac{1}{4}x\), apply the following transformations in order:

1. Stretch the graph horizontally by a factor of 4 in the x-direction. This changes \(\sin x\) to \(\sin \frac{1}{4}x\).

2. Stretch the graph vertically by a factor of 2 in the y-direction. This changes \(\sin \frac{1}{4}x\) to \(2 \sin \frac{1}{4}x\).

3. Translate the graph vertically by 3 units upwards. This changes \(2 \sin \frac{1}{4}x\) to \(3 + 2 \sin \frac{1}{4}x\).

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.

The original curve is \(y = \sin 2x - 5x\).

First, reflect the curve in the \(y\)-axis. This changes \(x\) to \(-x\), so the equation becomes \(y = \sin(-2x) - 5(-x) = -\sin 2x + 5x\).

Next, stretch the curve by a scale factor of \(\frac{1}{3}\) in the \(x\)-direction. This changes \(x\) to \(\frac{x}{\frac{1}{3}} = 3x\), so the equation becomes \(y = -\sin(2 \cdot 3x) + 5(3x) = -\sin 6x + 15x\).

(d) To map \(y = f(x)\) to \(y = g(x)\):

1. First, apply a horizontal stretch by a scale factor of \(\frac{1}{2}\) because \(g(x) = f(2x)\).
2. Then, translate the graph vertically by 4 units upwards, as indicated by \(g(x) = f(2x) + 4\).

(e) To map \(y = f(x)\) to \(y = h(x)\):

1. First, translate the graph horizontally by \(-\frac{\pi}{2}\) units, as indicated by \(h(x) = 2f\left(x + \frac{1}{2}\pi\right)\).
2. Then, apply a vertical stretch by a scale factor of 2, as indicated by the multiplication by 2 in \(h(x) = 2f\left(x + \frac{1}{2}\pi\right)\).