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

(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 \(y = f(x+4) - 3\).

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

(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.