To find the vertex of the quadratic function [tex]\( f(x) = x^2 - 8x - 9 \)[/tex], we can use the vertex formula. The vertex form for any quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by the coordinates [tex]\( \left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right) \)[/tex].
### Step-by-Step Solution:
1. Identify the coefficients:
- Here, the quadratic function is [tex]\( f(x) = x^2 - 8x - 9 \)[/tex].
- Comparing it with the standard form [tex]\( ax^2 + bx + c \)[/tex], we get:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = -9 \)[/tex]
2. Calculate the x-coordinate of the vertex:
- The x-coordinate of the vertex is given by [tex]\( \frac{-b}{2a} \)[/tex].
- Substituting [tex]\( b = -8 \)[/tex] and [tex]\( a = 1 \)[/tex] into the formula:
[tex]\[
x = \frac{-(-8)}{2 \cdot 1} = \frac{8}{2} = 4
\][/tex]
- Therefore, the x-coordinate of the vertex is [tex]\( x = 4 \)[/tex].
3. Calculate the y-coordinate of the vertex:
- To find the y-coordinate, we need to substitute [tex]\( x = 4 \)[/tex] back into the original quadratic function [tex]\( f(x) \)[/tex].
- Evaluating [tex]\( f(4) \)[/tex]:
[tex]\[
f(4) = (4)^2 - 8(4) - 9
\][/tex]
[tex]\[
f(4) = 16 - 32 - 9
\][/tex]
[tex]\[
f(4) = -25
\][/tex]
- Therefore, the y-coordinate of the vertex is [tex]\( y = -25 \)[/tex].
### Conclusion:
- The vertex of the quadratic function [tex]\( f(x) = x^2 - 8x - 9 \)[/tex] is at the point [tex]\( (4, -25) \)[/tex].
Thus, the vertex is [tex]\((4.0, -25.0)\)[/tex].
