<p>For a quadratic, \(a\ne0\). Each form reveals a different graph feature.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 580px;"><colgroup><col style="width: 20.6897%;"><col style="width: 25.8621%;"><col style="width: 25.8621%;"><col style="width: 27.5862%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Form</strong></th><th style="text-align: center; --cell-justify: center"><strong>General equation</strong></th><th style="text-align: center; --cell-justify: center"><strong>Feature it reveals</strong></th><th style="text-align: center; --cell-justify: center"><strong>Example</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Expanded</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Factorised</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Completed square</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>\(x^2-2x-3=(x+1)(x-3)=(x-1)^2-4\)</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" data-proportional style="width:300px;height:200px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:300px;--graph-aspect:300 / 200;"></div>
<p class="paragraph-header">Line of symmetry</p>
<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">\(x=-\dfrac{b}{2a}\)</p></div></div>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 37.0370%;"><col style="width: 62.9630%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Form you have</strong></th><th style="text-align: center; --cell-justify: center"><strong>Line of symmetry</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(y=ax^2+bx+c\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(y=a(x-p)(x-q)\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(y=a(x-h)^2+k\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>Set each factor equal to zero: \((x+1)=0\) gives&nbsp;\(x=-1\).</p>

For a quadratic, \(a\ne0\). Each form reveals a different graph feature.

\(y=x^2-2x-3\)
\(y\)-intercept \((0,-3)\)

\(\begin{aligned} y&=(x+1)(x-3) \\ x&=-1,\ x=3 \end{aligned}\)

\(x=\dfrac{p+q}{2}\) — midway between the \(x\)-intercepts

Set each factor equal to zero: \((x+1)=0\) gives \(x=-1\).

Form	General equation	Feature it reveals	Example
Expanded
Factorised
Completed square

Three Forms of a Quadratic Function

Three Forms of a Quadratic Function

Form you have	Line of symmetry
\(y=ax^2+bx+c\)
\(y=a(x-p)(x-q)\)
\(y=a(x-h)^2+k\)