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vertex_of_a_parabola.html
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vertex_of_a_parabola.html
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<!DOCTYPE html>
<html data-require="math math-format graphie interactive ast math-model subhints">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Vertex of a parabola</title>
<script src="../khan-exercise.js"></script>
<style type="text/css">
#answer_area input[type=text] {
width: 30px;
}
</style>
</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>
<var id="GROUP1">[
parse( "y &= A( x - #{h})^2 + #{k}", [ GREEN, GREEN ] ),
parse( "y &= " + A_DISP + "( x - #{" + H + "})^2 + #{" + K + "}", [ GREEN, GREEN ] )
]</var>
<var id="COMP_SQR1">[
parse( "y &= " + plus( A + "x^2", ( -2 * A * H ) + "x", ( A * H * H + K ) ) ),
parse( "y + " + ( -A * H * H - K ) + " &= " + plus( A + "x^2", ( -2 * A * H ) + "x" ) ),
]</var>
<var id="COMP_SQR2">[
parse( plus( "y", ( -A * H * H - K ) ) + " = " + A_DISP + "(" + plus( "x^2", ( -2 * H ) + "x" ) + ")" ),
]</var>
<var id="COMP_SQR3">[
parse( plus( "y", ( -A * H * H - K ) ) + " &= " + A_DISP + "(" + plus( "x^2", ( -2 * H ) + "x" ) + ")" ),
parse( plus( "y", ( -A * H * H - K ) ) + " + #{" + ( A * H * H ) + "} &= " + A_DISP + "(" + plus( "x^2", ( -2 * H ) + "x" ) + " + #{" + ( H * H ) + "})", [ BLUE, BLUE ] ),
parse( plus( "y", ( ( -A * H * H - K ) + ( A * H * H ) ) ) + " &= " + A_DISP + "(" + plus( "x^2", ( -2 * H ) + "x", ( H * H ) ) + ")" ),
]</var>
<var id="COMP_SQR4">[
parse( plus( "y", -K ) + " = " + A_DISP + "(" + plus( "x", -H ) + ")^2" ),
]</var>
<var id="COMP_SQR5">[
parse( "y = " + A_DISP + "(x - " + H + ")^2 + " + K )
]</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>\large{\left(\right.}</code><span class="sol"><var>H</var></span><code>,\text{ }</code><span class="sol"><var>K</var></span><code>\large{\left.\right)}</code></p>
<div class="example">a point, like <code>(-1, 2)</code> </div>
</div>
<div class="hints">
<div>
<p>When the equation is rewritten in vertex form like this, the vertex is the point <code class="hint_green">(h, k)</code>:
<p><code>\qquad <var>formatGroup( GROUP1, [ 0 ] )</var></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 <var>formatGroup( COMP_SQR1, [ 0, 1 ] )</var></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 <var>formatGroup( COMP_SQR2, [ 0 ] )</var></code></p>
</div>
<p>
We can complete the square by taking half of the coefficient of our x term, squaring it, and adding it to both sides of the equation.
The coefficient of our x 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 class="hint_blue"><var>H * H</var></code>. <span data-if="A !== 1">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 class="hint_blue"><var>A * H * H</var></code>
to the left side to make sure we're adding the same thing to both sides.</span>
</p>
<p><code>\qquad <var>formatGroup( COMP_SQR3, [ 0, 1, 2 ] )</var></code></p>
<div>
<p>Now we can rewrite the expression in parentheses as a squared term:</p>
<p><code>\qquad <var>formatGroup( COMP_SQR4, [ 0 ] )</var></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 <var>formatGroup( COMP_SQR5, [ 0 ] )</var></code></p>
</div>
<div>
<p>Now that the equation is written in vertex form, the vertex is the point <code class="hint_green">(h, k)</code>:</p>
<p><code>\qquad <var>formatGroup( GROUP1, [ 0 ] )</var></code></p>
</div>
<p class="final_answer">
The vertex is <code>(<var>H</var>, <var>K</var>)</code>. Be sure to pay attention to the signs when interpreting an equation in vertex form.
</p>
</div>
</div>
</div>
</div>
</body>
</html>