The transformation from \(y = f(x)\) to \(y = f\left(\frac{1}{2}x\right)\) involves a horizontal stretch. Specifically, it is a stretch by a factor of 2 in the x-direction, as the x-values are effectively halved, making the graph wider.

The transformation from \(y = f(x)\) to \(y = 1 + f(x)\) involves a vertical translation. Specifically, it is a translation of 1 unit upwards in the y-direction, as 1 is added to the function value.

Therefore, the two transformations are: a stretch by a factor of 2 in the x-direction and a translation of 1 unit in the y-direction.

(a) Start with the equation \(y = x^2\). First, apply the reflection in the x-axis: \(y = -x^2\). Then apply the translation \(\begin{pmatrix} 3 \\ -1 \end{pmatrix}\), which shifts the graph 3 units to the right and 1 unit down. The equation becomes \(y = -(x-3)^2 - 1\).

(b) Start with the equation \(y = x^2\). First, apply the translation \(\begin{pmatrix} 3 \\ -1 \end{pmatrix}\), which shifts the graph 3 units to the right and 1 unit down: \(y = (x-3)^2 - 1\). Then apply the reflection in the x-axis: \(y = -(x-3)^2 + 1\).

(c) To map \(y = g(x)\) onto \(y = h(x)\), observe that \(y = g(x)\) is \(-(x-3)^2 - 1\) and \(y = h(x)\) is \(-(x-3)^2 + 1\). The transformation is a vertical translation of 2 units up, represented by \(\begin{pmatrix} 0 \\ 2 \end{pmatrix}\).

The transformation from \(y = f(x)\) to \(y = g(x)\) involves two steps:

1. **Translation**: The graph is translated vertically downwards by 2 units. This is represented by the vector \(\begin{pmatrix} 0 \\ -2 \end{pmatrix}\).

2. **Stretch**: The graph is stretched parallel to the x-axis with a scale factor of 2. This means that each x-coordinate is multiplied by 2, effectively widening the graph horizontally.

These transformations can be applied in either order to achieve the graph of \(y = g(x)\).

To transform \(y = f(x)\) to \(y = g(x)\), we perform the following transformations:

1. Stretch the graph of \(y = f(x)\) by a factor of 2 in the y-direction. This means that every y-coordinate of the graph is multiplied by 2.

2. Translate the resulting graph by \(\begin{pmatrix} -6 \\ 0 \end{pmatrix}\), which means shifting the graph 6 units to the left.

These transformations can be applied in either order to achieve the same result.

Start with the function \(f(x) = x^2 - 2x + 5\).

1. Stretch parallel to the x-axis with scale factor \(\frac{1}{2}\): Replace \(x\) with \(2x\). The function becomes \((2x)^2 - 2(2x) + 5 = 4x^2 - 4x + 5\).

2. Reflection in the y-axis: Replace \(x\) with \(-x\). The function becomes \((-2x)^2 - 2(-2x) + 5 = 4x^2 + 4x + 5\).

3. Stretch parallel to the y-axis with scale factor 3: Multiply the entire function by 3. The function becomes \(3(4x^2 + 4x + 5) = 12x^2 + 12x + 15\).

Thus, \(g(x) = 12x^2 + 12x + 15\).