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<!DOCTYPE html>
<html data-require="math">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Vertex of a parabola</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="standard-form">
<div class="vars" data-ensure="A * H * H + K !== 0">
<var id="A">randRangeNonZero(-5, 5)</var>
<var id="H">randRangeNonZero(-5, 5)</var>
<var id="K">randRangeNonZero(-5, 5)</var>
<var id="A_DISP">A === 1 ? "" : A === -1 ? "-" : A</var>
</div>
<div class="problem">
<p>Given the equation:</p>
<p>
<code>\qquad y = <var>A_DISP</var>x^2 + <var>-2 * A * H</var>x + <var>A *H * H + K</var></code>
</p>
</div>
<p class="question">
Find the parabola's vertex.
</p>
<div class="solution" data-type="multiple">
<p>
<code>(</code>
<span class="sol short40"><var>H</var></span>
<code>,</code>
<span class="sol short40"><var>K</var></span>
<code>)</code>
</p>
</div>
<div class="hints">
<div>
<p>
When the equation is rewritten in vertex form like this,
the vertex is the point <code>(\green{h}, \blue{k})</code>:
</p><p><code>\qquad
y = A(x - \green{h})^2 + \blue{k}
</code></p>
</div>
<div>
<p>
We can rewrite the equation in vertex form by
completing the square. First, move the constant term to
the left side of the equation:
</p><p><code>\qquad
\begin{eqnarray}
y &amp;=&amp; <var>A_DISP</var>x^2 + <var>-2 * A * H</var>x + <var>A * H * H + K</var> \\ \\
y - <var>A * H * H + K</var> &amp;=&amp; <var>A_DISP</var>x^2 + <var>-2 * A * H</var>x
\end{eqnarray}
</code></p>
</div>
<div data-if="A !== 1">
<p>
Next, we can factor out a <code><var>A</var></code>
from the right side:
</p><p><code>\qquad
y - <var>A * H * H + K</var> = <var>A</var>(x^2 + <var>-2 * H</var>x)
</code></p>
</div>
<p>
We can complete the square by taking half of the
coefficient of our <code>x</code> term, squaring it, and adding it
to both sides of the equation. The coefficient of our
<code>x</code> term is <code><var>-2 * H</var></code>, so half of it would be <code><var>-H</var></code>, and squaring
that gives us <code>\pink{<var>H * H</var>}</code>. Because we're adding the <code><var>H * H</var></code> inside
the parentheses on the right where it's being
multiplied by <code><var>A</var></code>, we need to add <code>\pink{<var>A * H * H</var>}</code> to the left side
to make sure we're adding the same thing to both sides.
</p>
<p><code>\qquad
\begin{eqnarray}
y - <var>A * H * H + K</var> &amp;=&amp; <var>A</var>(x^2 + <var>-2 * H</var>x) \\ \\
y - <var>A * H * H + K</var> + \pink{<var>A * H * H</var>} &amp;=&amp; <var>A</var>(x^2 + <var>-2 * H</var>x + \pink{<var>H * H</var>}) \\ \\
y - <var>K</var> &amp;=&amp; <var>A</var>(x^2 + <var>-2 * H</var>x + <var>H * H</var>)
\end{eqnarray}
</code></p>
<div>
<p>
Now we can rewrite the expression in parentheses as a
squared term:
</p><p><code>\qquad
y - <var>K</var> = <var>A</var>(x - <var>H</var>)^2
</code></p>
</div>
<div>
<p>Move the constant term to the right side of the equation. Now the equation is in vertex form:</p>
<p><code>\qquad
y = <var>A</var>(x - <var>H</var>)^2 + <var>K</var>
</code></p>
</div>
<div>
<p>
Now that the equation is written in vertex form, the
vertex is the point <code>(\green{h}, \blue{k})</code>:
</p><p><code>\qquad
y = A(x - \green{h})^2 + \blue{k}
</code></p>
</div>
<div>
<p><code>\qquad
y = <var>A</var>(x - \green{(<var>H</var>)})^2 + \blue{(<var>K</var>)}
</code></p><p>
The vertex is
<code>(\green{<var>H</var>}, \blue{<var>K</var>})</code>.
Be sure to pay attention to the signs when interpreting
an equation in vertex form.
</p>
</div>
</div>
</div>
</div>
</div>
</body>
</html>