View source for Vertex equations of a quadratic function and it's inverse

A parabola can be uniquely defined by its vertex <math>V=(v_x, v_y)</math> and one more point <math>P=(p_x, p_y)</math>.
The function term of the parabola then has the form
 
:<math>y = a (x-v_x)^2 + v_y.</math>

<math>a</math> can be determined by solving

:<math>p_y = a (p_x-v_x)^2 + v_y</math> for <math>a</math> which gives

:<math> a = (p_y - v_y) / (p_x - v_x)^2 .</math>


<jsxgraph width="300" height="300" box="box1">
(function() {
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();
             }]);

})();
</jsxgraph>
=== JavaScript code ===
<source lang="javascript">
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();
             }]);

})();
</source>

===Inverse quadratic function===
Conversely, also the inverse quadratic function can be uniquely defined by its vertex <math>V</math> and one more point <math>P</math>.
The function term of the inverse function has the form
 
:<math>y = \sqrt{(x-v_x)/a} + v_y.</math>

<math>a</math> can be determined by solving

:<math>p_y = \sqrt{(p_x-v_x)/a} + v_y</math> for <math>a</math> which gives

:<math>a = (p_x - v_x) / (p_y - v_y)^2.</math>


<jsxgraph width="300" height="300" box="box2">
(function() {
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();
             }]);

})();
</jsxgraph>
=== JavaScript code ===
<source lang="javascript">
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();
             }]);
</source>


[[Category:Examples]]
[[Category:Calculus]]
[[Category:Interpolation]]