To find the vertex of the parabola given by the equation:
[tex]\[
x^2 + 14x - y - 14 = 0
\][/tex]
we first need to rewrite it in the standard form of a quadratic equation, [tex]\( y = ax^2 + bx + c \)[/tex].
Step-by-Step Solution:
1. Rewrite the equation:
[tex]\[
x^2 + 14x - y - 14 = 0
\][/tex]
Rearrange it to isolate [tex]\( y \)[/tex]:
[tex]\[
y = x^2 + 14x - 14
\][/tex]
2. Identify the coefficients:
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 14 \)[/tex], and [tex]\( c = -14 \)[/tex].
3. Find the x-coordinate of the vertex:
The x-coordinate of the vertex for a parabola given by [tex]\( y = ax^2 + bx + c \)[/tex] is calculated using the formula:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Plug in [tex]\( a = 1 \)[/tex] and [tex]\( b = 14 \)[/tex]:
[tex]\[
x = -\frac{14}{2 \times 1} = -\frac{14}{2} = -7
\][/tex]
4. Find the y-coordinate of the vertex:
Substitute the x-coordinate back into the equation [tex]\( y = x^2 + 14x - 14 \)[/tex]:
[tex]\[
y = (-7)^2 + 14(-7) - 14
\][/tex]
Calculate step by step:
[tex]\[
y = 49 - 98 - 14
\][/tex]
[tex]\[
y = 49 - 112
\][/tex]
[tex]\[
y = -63
\][/tex]
5. Write the coordinates of the vertex:
[tex]\[
(x, y) = (-7, -63)
\][/tex]
Hence, the vertex of the parabola is:
[tex]\[
([-7], [-63])
\][/tex]
