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<title>Equation of a parabola</title>
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<div class="vars" data-ensure="( A !== 0 ) && -10 < C && C < 10 && B !== 0 ">
<var id="A"> randRange( -5, 5 ) </var>
<var id="H"> randRange( -5, 5 )</var>
<var id="K"> randRange( -5, 5 )</var>
<!-- y = A*(x-H)^2 + K = A(x^2-2Hx+H^2) + K = Ax^2 -2AHx + A*H^2+K -->
<var id="B">-2 * A * H</var>
<var id="C">A * H * H + K</var>
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<p>The equation of a parabola <code>P</code> is <code>y = <var>A</var>x^2 + <var>B</var>x + <var>C</var></code>.</p>
<p>What are its vertex <code>(h, k)</code> and its <code>y</code>-intercept?</p>
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<p class="short"><code>(h, k) = (</code><span class="sol"><var>H</var></span><code>,</code> <span class="sol"><var>K</var></span><code>)</code></p>
<p><code>y</code>-intercept <code>=</code> <span class="sol"><var>C</var></span></p>
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<p>The <code>y</code>-intercept is the point on the <code>y</code>-axis where <code>x = 0</code>.</p>
<p>If <code>x = 0</code>, we have <code>y = <var>A</var> \cdot 0^2 + <var>B</var> \cdot 0 + <var>C</var> = <var>C</var></code>, so the <code>y</code>-intercept is <code><var>C</var></code>.</p>
<p>The equation of a parabola with vertex <code>(h, k)</code> is <code>y = a(x - h)^2 + k</code>.</p>
<p>We can rewrite the given equation as <code><var>A</var>x^2 + <var>A</var> \cdot <var>- 2 * H</var>x + <var>H * A * H</var> + <var>K</var></code>, in order to get the form <code>a(x - h)^2</code></p>
i <p>We factor out <code>A</code>, giving <code>y = <var>A</var> ( x^2 + <var>2 * H</var>x + <var>H * H</var> ) + <var>K</var></code></p>
<p>The equation in the parentheses is of the form <code>( a + b )^2</code>, because <code> ( x^2 + <var>2 * H</var> + <var>H * H</var> ) = ( x^2 + 2 \cdot <var>H</var> + <var>H</var>^2 ) </code></p>
<p>Therefore, <code>y = <var>A</var>( x - <var>H</var>)^2 + <var>K</var></code>.</p>
<p data-if="H < 0"> <code> y = <var>A</var>(x - (<var>H</var>))^2 + <var>K</var></code>.</p>
<p>Thus, the center <code>(h, k)</code> is <code>(<var>H</var>, <var>K</var>)</code></p>
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