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 ''V=(v_x, v_y)'' and one more point ''P=(p_x, p_y)''.
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 (x-v_x)^2 + v_y.</math>
:<math>y = a (x-v_x)^2 + v_y.</math>


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


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


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




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


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


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


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





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();
             }]);