(a) The stretch in the y-direction changes the distance from the center to the maximum point from 2 to 6 (since the maximum point is at 12 and the minimum at 0, the center is at 6). Therefore, the scale factor of the stretch is \(\frac{6}{2} = 3\).

(b) The radius of the original circle is the distance from the center to the maximum point before the stretch, which is 2.

(c) After the translation \(\begin{pmatrix} 7 \\ -3 \end{pmatrix}\), the center of the circle moves from (1, 5) to \((1+7, 5-3) = (8, 2)\).

(d) The original center of the circle is found by reversing the translation from (8, 2): \((8-7, 2+3) = (1, 5)\).

To find \(g(x)\) in terms of \(f(x)\), we need to apply the given transformations to \(f(x)\).

1. The first transformation is a stretch in the \(x\)-direction by a factor of 0.5. This means we replace \(x\) with \(2x\) in the function \(f(x)\). So, the function becomes \(f(2x)\).

2. The second transformation is a translation by \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\), which means we add 1 to the entire function. Thus, the function becomes \(f(2x) + 1\).

Therefore, the expression for \(g(x)\) in terms of \(f(x)\) is \(g(x) = f(2x) + 1\).

(a) To express \(f(x) = x^2 - 4x + 9\) in the form \((x-a)^2 + b\), complete the square:

\(f(x) = (x-2)^2 + 5\)

(b) To express \(g(x) = 2x^2 + 4x + 12\) in the form \(2[(x+c)^2 + d]\), first factor out the 2:

\(g(x) = 2(x^2 + 2x + 6)\)

Complete the square for \(x^2 + 2x + 6\):

\(x^2 + 2x + 6 = (x+1)^2 + 5\)

Thus, \(g(x) = 2[(x+1)^2 + 5]\)

(c) Express \(g(x)\) in the form \(kf(x+h)\):

\(g(x) = 2(x^2 + 2x + 6) = 2[(x+3)^2 + 5]\)

So, \(g(x) = 2f(x+3)\) where \(k=2\) and \(h=3\)

(d) The transformations from \(y = f(x)\) to \(y = g(x)\) are:

- Translation by \(\begin{pmatrix} -3 \\ 0 \end{pmatrix}\)

- Stretch in the y-direction by a factor of 2

(a) To translate the curve \(y = x^2 + 2x - 5\) by \(\begin{pmatrix} -1 \\ 3 \end{pmatrix}\), we substitute \(x + 1\) for \(x\) in the equation and add 3 to the result:

\(y = (x+1)^2 + 2(x+1) - 5 + 3\)

Expanding, we get:

\(y = (x^2 + 2x + 1) + (2x + 2) - 5 + 3\)

\(y = x^2 + 4x + 1\)

(b) The transformation from \(y = x^2 + 2x - 5\) to \(y = 4x^2 + 4x - 5\) is a horizontal stretch. The coefficient of \(x^2\) changes from 1 to 4, indicating a stretch in the x-direction by a factor of \(\frac{1}{2}\), keeping the y-axis invariant.