(a) To find the minimum point, complete the square for \(y = x^2 - 8x + 5\):

\(y = (x-4)^2 - 16 + 5 = (x-4)^2 - 11\).

The minimum point is at \(x = 4\), giving \(y = -11\). Thus, the coordinates are \((4, -11)\).

(b) The curve is stretched by a factor of 2 parallel to the y-axis, so the y-coordinate of the minimum point becomes \(-11 \times 2 = -22\). Then, the curve is translated by \(\begin{pmatrix} 4 \\ 1 \end{pmatrix}\), so the x-coordinate becomes \(4 + 4 = 8\) and the y-coordinate becomes \(-22 + 1 = -21\). Thus, the coordinates of the minimum point of the transformed curve are \((8, -21)\).

(c) The equation of the transformed curve is found by applying the transformations to the original equation:

First, stretch by a factor of 2: \(y = 2(x^2 - 8x + 5) = 2x^2 - 16x + 10\).

Then, translate by \(\begin{pmatrix} 4 \\ 1 \end{pmatrix}\):

Replace \(x\) with \(x - 4\): \(y = 2((x-4)^2 - 11) + 1\).

Expand: \(y = 2(x^2 - 8x + 16 - 11) + 1 = 2x^2 - 16x + 10 + 1 = 2x^2 - 16x + 11\).

Thus, the equation of the transformed curve is \(y = 2x^2 - 32x + 107\).

(a) To express \(2x^2 - 8x + 14\) in the form \(2[(x-a)^2 + b]\), we complete the square for the quadratic expression:

Start with \(x^2 - 4x\). Complete the square:

\(x^2 - 4x = (x-2)^2 - 4\).

Thus, \(2x^2 - 8x = 2((x-2)^2 - 4) = 2(x-2)^2 - 8\).

Now, add 14: \(2(x-2)^2 - 8 + 14 = 2(x-2)^2 + 6\).

Therefore, \(2x^2 - 8x + 14 = 2[(x-2)^2 + 3]\).

(b) To map \(y = f(x) = x^2\) to \(y = g(x) = 2x^2 - 8x + 14\), we perform the following transformations:

1. Translate by \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\): This shifts the graph right by 2 units and up by 3 units.

OR

1. Stretch in the \(y\) direction by a factor of 2: This scales the graph vertically.

The transformation from \(y = f(x)\) to \(y = 3 - f(x)\) involves two steps:

1. Reflection in the x-axis: This changes \(y = f(x)\) to \(y = -f(x)\).

2. Translation 3 units in the positive y-direction: This changes \(y = -f(x)\) to \(y = 3 - f(x)\).

Alternatively, the transformations can be performed in reverse order:

1. Translation 3 units in the negative y-direction: This changes \(y = f(x)\) to \(y = f(x) - 3\).

2. Reflection in the x-axis: This changes \(y = f(x) - 3\) to \(y = 3 - f(x)\).

(a) The transformation from \(y = f(x)\) to \(y = f(2x)\) involves a horizontal compression by a factor of \(\frac{1}{2}\). This is because the function \(f(2x)\) compresses the x-values by a factor of 2, effectively scaling the x-direction by \(\frac{1}{2}\).

The transformation from \(y = f(2x)\) to \(y = f(2x) - 3\) involves a vertical translation downwards by 3 units. This is because subtracting 3 from the function shifts the graph down by 3 units.

(b) To find the corresponding point on the original curve \(y = f(x)\), we reverse the transformations. The point \(P(5, 6)\) on \(y = f(2x) - 3\) corresponds to \(y = f(2x)\) at \((5, 9)\) after reversing the vertical translation.

Next, reverse the horizontal compression by multiplying the x-coordinate by 2, giving \((10, 9)\) on \(y = f(x)\).

(a) To express \(f(x) = -3x^2 + 12x + 2\) in the form \(-3(x-a)^2 + b\), we complete the square:

Start with \(-3(x^2 - 4x) + 2\).

Complete the square for \(x^2 - 4x\):

\(x^2 - 4x = (x-2)^2 - 4\).

Substitute back:

\(-3((x-2)^2 - 4) + 2 = -3(x-2)^2 + 12 + 2 = -3(x-2)^2 + 14\).

Thus, \(a = 2\) and \(b = 14\).

(e) Given the translation vector \(\begin{pmatrix} -3 \\ 1 \end{pmatrix}\), the function \(g(x)\) is obtained by substituting \(x+3\) into \(f(x)\) and adding 1:

\(g(x) = -3((x+3)-2)^2 + 14 + 1\).

Simplify:

\(g(x) = -3(x+1)^2 + 15\).

Expand \(-3(x+1)^2\):

\(-3(x^2 + 2x + 1) = -3x^2 - 6x - 3\).

Thus, \(g(x) = -3x^2 - 6x - 3 + 15 = -3x^2 - 6x + 12\).