Quadratic graph explorer


<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center">Quadratic function</th><th style="text-align: center; --cell-justify: center">Graph</th><th style="text-align: center; --cell-justify: center"><p><strong>Roots</strong></p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>\(y = a(x - h)^2 - k \)</p></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjE1MCwiaGVpZ2h0IjoxMDAsImVsZW1lbnRzIjpbeyJpZCI6IjhmMzVhOGFhLWE1NzEtNGZiMS04YzA2LWU4NWUxODczNWEyNiIsInR5cGUiOiJmdW5jdGlvbmdyYXBoIiwiY29sb3IiOiIjMDAwMDAwIiwieU9mZnNldCI6MCwiZXhwcmVzc2lvbiI6InheMiAtIDYqeCArIDgiLCJzdHJva2VXaWR0aCI6MS41fSx7IngiOjMsInkiOi0xLCJpZCI6ImI3N2Q2YzYxLWE4MTUtNDk4NC1hYTVmLWE2Yjk4ZWUyYTE0OSIsInR5cGUiOiJwb2ludCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoKGgsLWspXFwpIiwicG9pbnRTaXplIjoxLCJsYWJlbE9mZnNldCI6WzAsLTEwXX1dLCJ4QXhpc0xhYmVsIjoiXFwoeFxcKSIsInlBeGlzTGFiZWwiOiJcXCh5XFwpIiwiYm91bmRpbmdCb3giOlstMywxNC41LDEwLjUsLTMuNV0sIm9yaWdpbkxhYmVsIjoiXFwoT1xcKSIsImdyaWRTZXR0aW5ncyI6eyJtYWpvckNvbG9yIjoiIzk0YTNiOCIsIm1ham9yU3RlcFgiOjEsIm1ham9yU3RlcFkiOjEsIm1pbm9yQ29sb3IiOiIjY2JkNWUxIiwibWFqb3JPcGFjaXR5IjowLjQ1LCJtaW5vck9wYWNpdHkiOjAuNDUsIm1pbm9yRWxlbWVudHNYIjoxLCJtaW5vckVsZW1lbnRzWSI6MSwibWFqb3JTdHJva2VXaWR0aCI6MC44LCJtaW5vclN0cm9rZVdpZHRoIjowLjV9LCJzaG93QXhpc0xhYmVscyI6ZmFsc2UsInNob3dOYXZpZ2F0aW9uIjpmYWxzZSwia2VlcEFzcGVjdFJhdGlvIjpmYWxzZSwieEF4aXNMYWJlbE9mZnNldCI6WzIsLTExXSwieUF4aXNMYWJlbE9mZnNldCI6Wy0xNCw1XSwib3JpZ2luTGFiZWxPZmZzZXQiOlstNSwtMl19" data-width="150" data-height="100" data-proportional style="width:150px;height:100px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:150px;--graph-aspect:150 / 100;"></div></td><td style="text-align: center; --cell-justify: center"><p>\( h\pm\sqrt{\frac{k}{a}}\)<br></p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>\(y = -a(x - h)^2 + k \)</p></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="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" data-width="150" data-height="100" data-proportional style="width:150px;height:100px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:150px;--graph-aspect:150 / 100;"></div></td><td style="text-align: center; --cell-justify: center"><p>\( h\pm\sqrt{\frac{k}{a}}\)</p></td></tr></tbody></table></div></div>
<p>Take \(a\ne0\). The displayed roots are real when&nbsp;\(k/a\ge0\); they coincide when&nbsp;\(k=0\). In the illustrated upward/downward cases, \(a&gt;0\).</p>

Take \(a\ne0\). The displayed roots are real when \(k/a\ge0\); they coincide when \(k=0\). In the illustrated upward/downward cases, \(a>0\).

Quadratic Functions in the form of \(y=a{{\left( x-h \right)}^{2}}+k\)

Quadratic Functions in the form of \(y=a{{\left( x-h \right)}^{2}}+k\)

Quadratic function	Graph	Roots
\(y = a(x - h)^2 - k \)		\( h\pm\sqrt{\frac{k}{a}}\)
\(y = -a(x - h)^2 + k \)		\( h\pm\sqrt{\frac{k}{a}}\)