To determine the vertex of the quadratic function [tex]\( f(x) = x^2 - 2x - 3 \)[/tex], we need to follow these steps:
1. Identify the coefficients: For the quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
- [tex]\( c = -3 \)[/tex]
2. Find the x-coordinate of the vertex using the vertex formula [tex]\( x = -\frac{b}{2a} \)[/tex]:
[tex]\[
x = -\frac{-2}{2(1)} = \frac{2}{2} = 1
\][/tex]
3. Substitute the x-coordinate back into the original function to find the y-coordinate. The function is [tex]\( f(x) = x^2 - 2x - 3 \)[/tex]. Substituting [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4
\][/tex]
So, the vertex of the quadratic function [tex]\( f(x) = x^2 - 2x - 3 \)[/tex] is at the point [tex]\( (1, -4) \)[/tex].
Among the given choices, the correct answer is [tex]\((1, -4)\)[/tex].
