Vertex equations of a quadratic function and it's inverse

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

[math]\displaystyle{ y = a (x-v_x)^2 + v_y. }[/math]

[math]\displaystyle{ a }[/math] can be determined by solving

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

[math]\displaystyle{ a = (p_y - v_y) / (p_x - v_x)^2 . }[/math]

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();
             }]);

})();

Inverse quadratic function

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

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

[math]\displaystyle{ a }[/math] can be determined by solving

[math]\displaystyle{ p_y = \sqrt{(p_x-v_x)/a} + v_y }[/math] for [math]\displaystyle{ a }[/math] which gives

[math]\displaystyle{ a = (p_x - v_x) / (p_y - v_y)^2. }[/math]

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();
             }]);