To find the vertex of the quadratic function [tex]\( f(x) = x^2 - 8x - 9 \)[/tex], we need to determine the coordinates of the vertex using the vertex formula.
The general form of the quadratic function is [tex]\( f(x) = ax^2 + bx + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = -9 \)[/tex].
Step-by-Step Solution:
1. Determine the x-coordinate of the vertex:
The x-coordinate of the vertex for a quadratic function is given by the formula:
[tex]\[
x = \frac{-b}{2a}
\][/tex]
Substituting the given values [tex]\( a = 1 \)[/tex] and [tex]\( b = -8 \)[/tex]:
[tex]\[
x = \frac{-(-8)}{2 \cdot 1} = \frac{8}{2} = 4
\][/tex]
So the x-coordinate of the vertex is [tex]\( 4 \)[/tex].
2. Determine the y-coordinate of the vertex:
To find the y-coordinate, we substitute the x-coordinate back into the original quadratic function [tex]\( f(x) \)[/tex]:
[tex]\[
y = f(4) = (4)^2 - 8(4) - 9
\][/tex]
Calculate each term:
[tex]\[
4^2 = 16
\][/tex]
[tex]\[
-8 \cdot 4 = -32
\][/tex]
[tex]\[
(4)^2 - 8(4) - 9 = 16 - 32 - 9 = -25
\][/tex]
So the y-coordinate of the vertex is [tex]\(-25\)[/tex].
3. Write the coordinates of the vertex:
The coordinates of the vertex are:
[tex]\[
(x, y) = (4, -25)
\][/tex]
Thus, the vertex of the quadratic function [tex]\( f(x) = x^2 - 8x - 9 \)[/tex] is [tex]\((4, -25)\)[/tex].
