(a) To express \(f(x) = x^2 - 2x + 5\) in completed square form, we complete the square:

\(f(x) = (x-1)^2 + 4\).

For \(g(x) = x^2 + 4x + 13\), complete the square:

\(g(x) = (x+2)^2 + 9\).

Now express \(g(x)\) in the form \(f(x+p) + q\):

\(g(x) = (x+2)^2 + 9 = ((x+3)-1)^2 + 4 + 5 = f(x+3) + 5\).

Thus, \(p = 3\) and \(q = 5\).

(b) The transformation from \(y = f(x)\) to \(y = g(x)\) is a translation. The completed square forms show a horizontal shift by \(-3\) and a vertical shift by \(5\). Therefore, the transformation is a translation by \(\begin{pmatrix} -3 \\ 5 \end{pmatrix}\).

The transformation from \(y = f(x)\) to \(y = 2f(x - 1)\) involves two steps:

1. Translation: The expression \(f(x - 1)\) indicates a translation of the graph by 1 unit in the positive x-direction.

2. Stretch: The factor of 2 in \(2f(x - 1)\) indicates a vertical stretch by a factor of 2.

Therefore, the graph is translated 1 unit to the right and then stretched vertically by a factor of 2.

(a) The transformation involves two steps:

1. A stretch parallel to the \(y\)-axis by a factor of 3. This means every \(y\)-coordinate of the graph \(y = f(x)\) is multiplied by 3.

2. A translation 4 units in the positive \(x\)-direction. This shifts the graph horizontally to the right by 4 units.

(b) The equation of the transformed graph can be written as \(y = 3f(x-4)\). This accounts for the stretch by a factor of 3 and the horizontal translation by 4 units.

(a) To express \(x^2 + 6x + 5\) in the form \((x + a)^2 + b\), we complete the square:

Start with \(x^2 + 6x\). Take half of the coefficient of \(x\), which is 3, and square it to get 9.

Add and subtract 9 inside the expression: \(x^2 + 6x = (x^2 + 6x + 9) - 9 = (x + 3)^2 - 9\).

Now, include the constant term 5: \((x + 3)^2 - 9 + 5 = (x + 3)^2 - 4\).

Thus, \(x^2 + 6x + 5 = (x + 3)^2 - 4\).

(b) The transformation from \(y = x^2\) to \(y = x^2 + 6x + 5\) involves completing the square as shown in part (a): \(y = (x + 3)^2 - 4\).

This represents a translation of the graph of \(y = x^2\) by \(\begin{pmatrix} -3 \\ -4 \end{pmatrix}\), which means 3 units to the left and 4 units down.

(a) The transformation shown is a reflection in the y-axis. The equation of the transformed graph is \(y = f(-x)\).

(b) The transformation shown is a vertical stretch with a scale factor of 2. The equation of the transformed graph is \(y = 2f(x)\).

(c) The transformation shown is a translation 4 units to the left and 3 units down. The equation of the transformed graph is \(y = f(x+4) - 3\).