Quadratic graph explorer

<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(y=a(x-h)^2+k\quad(a\ne0)\)</p></div></div>
<p>The turning point is \((h,k)\). Factor out \(a\) from the quadratic and linear terms before completing the square.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 18.5185%;"><col style="width: 37.0370%;"><col style="width: 44.4444%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Sign of \(a\)</strong></th><th style="text-align: center; --cell-justify: center"><strong>Turning point</strong></th><th style="text-align: center; --cell-justify: center"><strong>Sketch</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(a&gt;0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="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" data-width="100" data-height="100" data-proportional style="width:100px;height:100px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:100px;--graph-aspect:100 / 100;"></div></td></tr><tr><td style="text-align: center; --cell-justify: center">\(a&lt;0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="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" data-width="100" data-height="100" data-proportional style="width:100px;height:100px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:100px;--graph-aspect:100 / 100;"></div></td></tr></tbody></table></div></div>

The turning point is \((h,k)\). Factor out \(a\) from the quadratic and linear terms before completing the square.

Sign of \(a\)	Turning point	Sketch
\(a>0\)
\(a<0\)

Maximum and Minimum Values by Completing the Square

Maximum and Minimum Values by Completing the Square