# Vertex equations of a quadratic function and it's inverse

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A parabola can be uniquely defined by its vertex $\displaystyle{ V=(v_x, v_y) }$ and one more point $\displaystyle{ P=(p_x, p_y) }$. The function term of the parabola then has the form

$\displaystyle{ y = a (x-v_x)^2 + v_y. }$

$\displaystyle{ a }$ can be determined by solving

$\displaystyle{ p_y = a (p_x-v_x)^2 + v_y }$ for $\displaystyle{ a }$ which gives
$\displaystyle{ a = (p_y - v_y) / (p_x - v_x)^2 . }$

### JavaScript code

var b = JXG.JSXGraph.initBoard('box1', {boundingbox: [-5, 5, 5, -5], grid:true});
var v = b.create('point', [0,0], {name:'V'}),
p = b.create('point', [3,3], {name:'P'}),
f = b.create('functiongraph', [
function(x) {
var den = p.X()- v.X(),
a = (p.Y() - v.Y()) / (den * den);
return a * (x - v.X()) * (x - v.X()) + v.Y();
}]);

})();

Conversely, also the inverse quadratic function can be uniquely defined by its vertex $\displaystyle{ V }$ and one more point $\displaystyle{ P }$. The function term of the inverse function has the form

$\displaystyle{ y = \sqrt{(x-v_x)/a} + v_y. }$

$\displaystyle{ a }$ can be determined by solving

$\displaystyle{ p_y = \sqrt{(p_x-v_x)/a} + v_y }$ for $\displaystyle{ a }$ which gives
$\displaystyle{ a = (p_x - v_x) / (p_y - v_y)^2. }$

### JavaScript code

var b = JXG.JSXGraph.initBoard('box2', {boundingbox: [-5, 5, 5, -5], grid:true});
var v = b.create('point', [0,0], {name:'V'}),
p = b.create('point', [3,3], {name:'P'}),
f = b.create('functiongraph', [
function(x) {
var den = p.Y()- v.Y(),
a = (p.X() - v.X()) / (den * den),
sign = (p.Y() >= 0) ? 1 : -1;
return sign * Math.sqrt((x - v.X()) / a) + v.Y();
}]);