Vertex equations of a quadratic function and it's inverse: Difference between revisions
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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"> | <jsxgraph width="300" height="300" box="box1"> | ||
(function() { | (function() { | ||
var b = JXG.JSXGraph.initBoard('box1', {boundingbox: [-5, 5, 5, -5], grid:true}); | var b = JXG.JSXGraph.initBoard('box1', {boundingbox: [-5, 5, 5, -5], grid:true}); | ||
var v = b.create('point', [0,0], {name:'V'}), | var v = b.create('point', [0,0], {name:'V'}), | ||
p = b.create('point', [3,3], { | p = b.create('point', [3,3], {name:'P'}), | ||
f = b.create('functiongraph', [ | f = b.create('functiongraph', [ | ||
function(x) { | function(x) { | ||
var den = p.X()- v.X(), | var den = p.X()- v.X(), | ||
a = (p.Y() - v.Y()) / (den * den); | a = (p.Y() - v.Y()) / (den * den); | ||
return a * (x - v.X()) + v.Y(); | return a * (x - v.X()) * (x - v.X()) + v.Y(); | ||
}]); | }]); | ||
})(); | })(); | ||
</jsxgraph> | </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:Examples]] | ||
[[Category:Calculus]] | [[Category:Calculus]] | ||
[[Category:Interpolation]] | [[Category:Interpolation]] |
Latest revision as of 14:18, 15 January 2021
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();
}]);