Answered

20. Which is the vertex of the quadratic function [tex]\( f(x) = x^2 - 2x - 3 \)[/tex]?

A. [tex]\((-1, -4)\)[/tex]
B. [tex]\((-1, 4)\)[/tex]
C. [tex]\((1, -4)\)[/tex]
D. [tex]\((1, 4)\)[/tex]



Answer :

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].