To find the vertex of the parabola described by the equation [tex]\( y^2 - 4x - 8y - 12 = 0 \)[/tex], we will complete the square. Here are the steps to arrive at the vertex form of the given equation:
1. Start with the given equation:
[tex]\[
y^2 - 4x - 8y - 12 = 0
\][/tex]
2. Rearrange the equation to focus on the [tex]\( y \)[/tex] terms for completing the square:
[tex]\[
y^2 - 8y - 4x - 12 = 0
\][/tex]
3. Group the [tex]\( y \)[/tex] terms together:
[tex]\[
y^2 - 8y = 4x + 12
\][/tex]
4. Complete the square for the [tex]\( y \)[/tex] terms:
- Take the coefficient of [tex]\( y \)[/tex], which is [tex]\(-8\)[/tex], halve it to get [tex]\(-4\)[/tex], and then square it to get [tex]\(16\)[/tex].
- Add and subtract this square inside the equation:
[tex]\[
y^2 - 8y + 16 - 16 = 4x + 12
\][/tex]
5. Simplify the left side by combining the perfect square trinomial and adjusting the constants on the right side:
[tex]\[
(y - 4)^2 - 16 = 4x + 12
\][/tex]
6. Add 16 to both sides to isolate the perfect square trinomial on the left:
[tex]\[
(y - 4)^2 = 4x + 28
\][/tex]
7. Reformat the equation to the standard vertex form of a parabola, [tex]\((y - k)^2 = 4p(x - h)\)[/tex]:
[tex]\[
(y - 4)^2 = 4(x + 7)
\][/tex]
In this form, the vertex [tex]\((h, k)\)[/tex] of the parabola can be identified directly. For the equation [tex]\((y - 4)^2 = 4(x + 7)\)[/tex]:
- [tex]\( h = -7 \)[/tex]
- [tex]\( k = 4 \)[/tex]
Thus, the vertex of the parabola is:
[tex]\[
(-7, 4)
\][/tex]
So, the completed square form reveals that the vertex of the parabola described by the equation [tex]\( y^2 - 4x - 8y - 12 = 0 \)[/tex] is [tex]\( (-7, 4) \)[/tex].
