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The image contains a mathematical problem asking to "Transform the equation of the parabola to the standard vertex form by completing the square." The equations provided are:

1. \( y = x^2 + 6x + 9 \)
2. \( y = x^2 - 6x + 9 \)
3. \( y = -y^2 + 6y - 8 \)
4. \( x = y^2 - 4y + 14 \)

Let's solve each equation by completing the square and transforming them into their respective vertex forms.

### 1. \( y = x^2 + 6x + 9 \)

The equation is already in a form that can be factored easily:
\[
y = (x + 3)^2
\]
So, the vertex form is:
\[
y = (x + 3)^2
\]
The vertex is at \((-3, 0)\).

### 2. \( y = x^2 - 6x + 9 \)

Similarly, this equation can be factored:
\[
y = (x - 3)^2
\]
So, the vertex form is:
\[
y = (x - 3)^2
\]
The vertex is at \((3, 0)\).

### 3. \( y = -y^2 + 6y - 8 \)

First, we rewrite the equation in terms of \( y \):
\[
-y^2 + 6y - 8 = 0
\]
Multiply the entire equation by -1:
\[
y^2 - 6y + 8 = 0
\]
Complete the square:
\[
y^2 - 6y + 9 = 1
\]
So, we have:
\[
(y - 3)^2 = 1
\]
Thus, the equation in vertex form is:
\[
x = (y - 3)^2 + 1
\]
The vertex is at \((1, 3)\).

### 4. \( x = y^2 - 4y + 14 \)

Complete the square for the \( y \) terms:
\[
x = (y^2 - 4y + 4) + 10
\]
\[
x = (y - 2)^2 + 10
\]
So, the vertex form is:
\[
x = (y - 2)^2 + 10
\]
The vertex is at \((10, 2)\).

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Do you want more details or have any questions?

### Related Questions:
1. How do you derive the vertex form from the standard form for any quadratic equation?
2. Can you explain the process of completing the square in more detail?
3. What is the significance of the vertex in a parabola's graph?
4. How can you determine if a parabola opens upwards or downwards by looking at its equation?
5. Can the method of completing the square be used for any quadratic equation?

### Tip:
When completing the square, remember to always balance the equation by adding and subtracting the same value inside the equation to maintain equality.

Learn how to transform parabola equations to vertex form by completing the square. Step-by-step solutions provided for four different equations, covering key algebraic concepts of completing the square and vertex form. Suitable for high school students.

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