# Difference between revisions of "Vertex equations of a quadratic function and it's inverse"

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− | A parabola can be uniquely defined by its vertex ''V'' and one more point ''P''. | + | A parabola can be uniquely defined by its vertex ''V=(v_x, v_y)'' and one more point ''P=(p_x, p_y)''. |

The function term of the parabola then has the form | The function term of the parabola then has the form | ||

− | + | ||

+ | ''y = a (x-v_x)^2 + v_y''. | ||

+ | |||

+ | ''a'' can be determined by solving | ||

+ | |||

+ | ''p_y = a (p_x-v_x)^2 + v_y'' for ''a'' which gives | ||

+ | |||

+ | '' a = (p_y - v_y) / (p_x - v_x)^2 ''. | ||

+ | |||

Line 32: | Line 40: | ||

})(); | })(); | ||

</source> | </source> | ||

+ | |||

+ | ===Inverse quadratic function=== | ||

+ | Conversely, also the inverse quadratic function can be uniquely defined by its vertex ''V'' and one more point ''P''. | ||

+ | The function term of the inverse function has the form | ||

+ | |||

+ | ''y = sqrt((x-v_x)/a) + v_y''. | ||

+ | |||

+ | ''a'' can be determined by solving | ||

+ | |||

+ | ''p_y = sqrt((p_x-v_x)/a) + v_y'' for ''a'' which gives | ||

+ | |||

+ | '' a = (p_x - v_x) / (p_y - v_y)^2 ''. | ||

+ | |||

<jsxgraph width="300" height="300" box="box2"> | <jsxgraph width="300" height="300" box="box2"> | ||

Line 41: | Line 62: | ||

function(x) { | function(x) { | ||

var den = p.Y()- v.Y(), | var den = p.Y()- v.Y(), | ||

− | a = (p.X() - v.X()) / (den * den); | + | a = (p.X() - v.X()) / (den * den), |

− | return Math.sqrt((x - v.X()) / a) + v.Y(); | + | sign = (p.Y() >= 0) ? 1 : -1; |

+ | return sign * Math.sqrt((x - v.X()) / a) + v.Y(); | ||

}]); | }]); | ||

Line 55: | Line 77: | ||

function(x) { | function(x) { | ||

var den = p.Y()- v.Y(), | var den = p.Y()- v.Y(), | ||

− | a = (p.X() - v.X()) / (den * den); | + | a = (p.X() - v.X()) / (den * den), |

− | return Math.sqrt((x - v.X()) / a) + v.Y(); | + | sign = (p.Y() >= 0) ? 1 : -1; |

+ | return sign * Math.sqrt((x - v.X()) / a) + v.Y(); | ||

}]); | }]); | ||

− | |||

</source> | </source> | ||

## Latest revision as of 15:21, 16 December 2014

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

*y = a (x-v_x)^2 + v_y*.

*a* can be determined by solving

*p_y = a (p_x-v_x)^2 + v_y* for *a* which gives

* 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();
}]);
})();
```

### Inverse quadratic function

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

*y = sqrt((x-v_x)/a) + v_y*.

*a* can be determined by solving

*p_y = sqrt((p_x-v_x)/a) + v_y* for *a* which gives

* 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();
}]);
```