Which of the following represents the vertex of the parabola described by y = ax² + bx + c?

The vertex of a parabola given by the quadratic equation ( y = ax^2 + bx + c ) can be found using the formula for the x-coordinate, which is ( -\frac{b}{2a} ). This formula comes from the process of completing the square or by using calculus to find the minimum or maximum point of the quadratic function.

After determining the x-coordinate of the vertex, it is essential to calculate the corresponding y-coordinate. This is done by substituting ( -\frac{b}{2a} ) back into the original equation ( f(x) ) to find ( f\left(-\frac{b}{2a}\right) ). Thus, the coordinates of the vertex are represented as ( \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) ).

With this understanding, it is clear why the first choice accurately reflects the vertex's coordinates. The x-coordinate is correctly derived, and the y-coordinate is found by evaluating the function at that x-coordinate, providing a complete representation of the vertex of the parabola. This aligns perfectly with the standard form and the mathematical principles governing quadratic equations.