Source: algebrica.org - CC BY-NC 4.0 https://algebrica.org/geometrical-meaning-quadratic-equations/ Fetched from algebrica.org post 525; source modified 2026-03-08T21:24:52.
From Equation to Parabola
The graphical representation of the related function $y = ax^2 + bx + c$, associated with a quadratic equation $ax^2 + bx + c = 0$, is a parabola. Its vertex corresponds to the maximum or minimum point of the curve, depending on the sign of the coefficient $a$: the vertex is a minimum if $a > 0$ and a maximum if $a < 0$. The shape and position of the parabola are determined by the values of the coefficients $a$, $b$, and $c$.
- The coefficient $a$ controls the direction, width, and steepness of the parabola: larger absolute values of $a$ make the graph narrower, while smaller values make it wider.
- The coefficient $b$ influences the horizontal position of the vertex.
- The constant term $c$ determines the vertical shift of the entire curve.
If the parabola is expressed in standard form as $f(x) = ax^2 + bx + c$, then:
If $a > 0$, the parabola opens upward $\cup$ and has a minimum point.
If $a < 0$, the parabola opens downward $\cap$ and has a maximum point.
In both cases, the coordinates of the vertex are given by:
If the parabola is expressed in the standard form $f(y) = ay^2 + by + c$, then:
If $a > 0$, the parabola opens to the right $\subset$.
If $a < 0$, the parabola opens to the left $\supset$.
In both cases, the vertex coordinates are given by:
Vertex and symmetry in special cases
Graphically, a generic $y = ax^2 + bx + c$ parabola with its axis parallel to the y-axis looks like the following:
When $b = 0$ and $c \neq 0$ the equation becomes $y = ax^2 + c$. The parabola has its vertex at $V(0, c)$, and its axis of symmetry is the y-axis.
When Case: $c = 0$ and $b \neq 0$ the equation becomes $y = ax^2 + bx$. The parabola has its vertex at:
The parabola always passes through the origin ( 0, 0 ).