To determine the correct formula for the quadratic function [tex]\( y = f(x) \)[/tex] given that its vertex is [tex]\( (2, -1) \)[/tex] and it passes through the point [tex]\( (3, 0) \)[/tex], we will follow these steps:
1. Vertex Form of a Quadratic Function:
The general vertex form of a quadratic function is given by:
[tex]\[
y = a(x - h)^2 + k
\][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola. Given the vertex [tex]\((2, -1)\)[/tex], our equation becomes:
[tex]\[
y = a(x - 2)^2 - 1
\][/tex]
2. Substitute the Given Point to Solve for [tex]\( a \)[/tex]:
We know that the quadratic function passes through the point [tex]\((3, 0)\)[/tex]. Substituting [tex]\( x = 3 \)[/tex] and [tex]\( y = 0 \)[/tex] into the equation:
[tex]\[
0 = a(3 - 2)^2 - 1
\][/tex]
Simplify the equation:
[tex]\[
0 = a(1)^2 - 1
\][/tex]
[tex]\[
0 = a - 1
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
a = 1
\][/tex]
3. Form the Final Quadratic Equation:
Substitute [tex]\( a = 1 \)[/tex] back into the vertex form equation:
[tex]\[
y = (x - 2)^2 - 1
\][/tex]
4. Identify the Correct Option:
Comparing the obtained equation [tex]\( y = (x - 2)^2 - 1 \)[/tex] with the provided options, we see that this corresponds to option B.
Thus, the formula for the quadratic function [tex]\( y = f(x) \)[/tex] is given by:
[tex]\[
\boxed{(x - 2)^2 - 1}
\][/tex]
Hence, the correct answer is option B.
