The task involves converting quadratic functions from their general form $y = ax^2 + bx + c$ into vertex form $y = a(x - h)^2 + k$, where $h$ and $k$ represent the coordinates of the vertex, and $a$ is the same as the leading coefficient in both forms.

Here are the steps to convert each equation to vertex form by completing the square:

1. $y = x^2 - 6x - 6$

• Start by focusing on $x^2 - 6x$.
• Take half of $-6$, square it, and add it inside the equation: $\left( \frac{-6}{2} \right)^2 = 9$.
• Add and subtract 9: $y = (x^2 - 6x + 9) - 9 - 6$.
• This simplifies to: $y = (x - 3)^2 - 15$.

Thus, the vertex form is:
$y = (x - 3)^2 - 15$ So, $a = 1$, $h = 3$, and $k = -15$.

2. $y = x^2 + 4x - 10$

• Focus on $x^2 + 4x$.
• Take half of 4, square it, and add it: $\left( \frac{4}{2} \right)^2 = 4$.
• Add and subtract 4: $y = (x^2 + 4x + 4) - 4 - 10$.
• Simplifies to: $y = (x + 2)^2 - 14$.

Thus, the vertex form is:
$y = (x + 2)^2 - 14$
So, $a = 1$, $h = -2$, and $k = -14$.

3. $y = 3x^2 - 18x + 2$

• Factor out 3 from the first two terms: $y = 3(x^2 - 6x) + 2$.
• Complete the square inside the parentheses: $\left( \frac{-6}{2} \right)^2 = 9$.
• Add and subtract $9 \times 3$ inside: $y = 3(x^2 - 6x + 9) - 27 + 2$.
• Simplifies to: $y = 3(x - 3)^2 - 25$.

Thus, the vertex form is:
$y = 3(x - 3)^2 - 25$
So, $a = 3$, $h = 3$, and $k = -25$.

4. $y = x^2 - 16x + 14$

• Focus on $x^2 - 16x$.
• Half of $-16$ is $-8$, and $\left( \frac{-16}{2} \right)^2 = 64$.
• Add and subtract 64: $y = (x^2 - 16x + 64) - 64 + 14$.
• Simplifies to: $y = (x - 8)^2 - 50$.

Thus, the vertex form is:
$y = (x - 8)^2 - 50$
So, $a = 1$, $h = 8$, and $k = -50$.

5. $y = 2x^2 + 8x - 13$

• Factor out 2 from the first two terms: $y = 2(x^2 + 4x) - 13$.
• Complete the square inside: $\left( \frac{4}{2} \right)^2 = 4$.
• Add and subtract $4 \times 2$: $y = 2(x^2 + 4x + 4) - 8 - 13$.
• Simplifies to: $y = 2(x + 2)^2 - 21$.

Thus, the vertex form is:
$y = 2(x + 2)^2 - 21$
So, $a = 2$, $h = -2$, and $k = -21$.

Would you like more details on any step? Here are some related questions to explore:

1. How is completing the square useful for solving quadratic equations?
2. Can vertex form help in graphing quadratic functions more easily?
3. What is the geometric significance of the constants $h$ and $k$?
4. How does factoring differ from completing the square in quadratic transformations?
5. What are some real-life applications of quadratic functions?

Tip: Always double-check your constants when completing the square to avoid mistakes.