Difference between revisions of "Vertex equations of a quadratic function and it's inverse"
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Revision as of 16:13, 15 January 2021
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 (xv_x)^2 + v_y.
a can be determined by solving
p_y = a (p_xv_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((xv_x)/a) + v_y.
a can be determined by solving
p_y = sqrt((p_xv_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();
}]);