Completing the square

Parametric equations

A quadratic ( $a \neq = 0$ ) can be written in general form

f (x) = a x^{2} + b x + c

or in vertex form

f (x) = a (x - h)^{2} + k

where the vertex is $V = (h, k)$ . The discriminant

Δ = b^{2} - 4 a c

decides whether the graph meets the $x$ -axis in two points, touches it at one point, or does not meet it in the real numbers.

Coefficient a, Coefficient b, and Coefficient c: drag $a$ , $b$ , and $c$ and watch the opening, the position, and the $x$ -intercepts
V: the vertex and the axis $x = h$ stay marked. Drag V on the figure and the coefficients $b$ and $c$ follow
Completing guide: on / Completing guide: off: show the base parabola $y = a x^{2}$ and, once the vertex has moved, the arrow from the origin to the vertex
Two real roots / Repeated root / No real roots: switch the three discriminant cases and compare the sign of $Δ$ with the $x$ -axis

$a$ sets the direction of the opening and how wide it is. A larger $∣ a ∣$ makes the parabola narrower.
The vertex form gives the shift directly. The general form is the one that computes $Δ$ and the intercepts from the coefficients.
$Δ > 0$ gives two intercepts, $Δ = 0$ means the graph touches the $x$ -axis, and $Δ < 0$ means the graph misses the $x$ -axis, so the equation has no real solution.