# Geometrical Meaning of Quadratic Equations

*Ref.*quadratic equations

The graphical representation of a quadratic equation \(ax^2 + bx + c = 0\) is a parabola, with its vertex corresponding to the minimum or maximum point of the equation, determined by the sign of the coefficient \(a\). The shape of the curve for a quadratic equation is determined by its coefficients.

The coefficient \(a\) controls the width and steepness of the curve, while \(b\) affects the horizontal position of the vertex. The constant term \(c\) shifts the entire curve vertically.

If the parabola is expressed in the standard form \(f(x) = ax^2+bx+c\) we have:

if \(a > 0\), the parabola opens upwards \(\cup\) with a minimum point.

If \(a <0 \), the parabola opens downwards \(\cap\) with a maximum point.

In this case the vertex coordinates are given by: \[ V = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) \]

We have: \(y = x^2 – 2x + 1\). \(a > 0\), the parabola opens upwards with a minimum point.

We have: \(y = -x^2 – 2x + 1\). \(a <0 \), the parabola opens downwards with a maximum point.

If the parabola is expressed in the standard form \(f(y) = ay^2+by+c\) we have:

If \(a > 0\), the parabola opens rightward \(\subset\).

If \(a <0 \), the parabola opens leftwards \(\supset\).

In this case the vertex coordinates are given by: \[ V = \left(f\left(-\frac{b}{2a}\right), -\frac{b}{2a}\right) \]