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''.
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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
 
The function term of the parabola then has the form
:<math>
+
    y  = a(v-v_x)^2 + v_y
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:<math>y = a (x-v_x)^2 + v_y.</math>
</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">
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})();
 
})();
 
</source>
 
</source>
 +
 +
===Inverse quadratic function===
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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">
 
<jsxgraph width="300" height="300" box="box2">
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             function(x) {
 
             function(x) {
 
                 var den = p.Y()- v.Y(),
 
                 var den = p.Y()- v.Y(),
                     a = (p.X() - v.X()) / (den * den);
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                     a = (p.X() - v.X()) / (den * den),
                 return Math.sqrt((x - v.X()) / a) + v.Y();
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                    sign = (p.Y() >= 0) ? 1 : -1;
 +
                 return sign * Math.sqrt((x - v.X()) / a) + v.Y();
 
             }]);
 
             }]);
  
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             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 16:18, 15 January 2021

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]


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]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]


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();
             }]);