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<!DOCTYPE html>
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<title>Hyperbolas</title>
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<div class="section" id="fwk-redden-ch08_s04" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">8.4</span> Hyperbolas</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch08_s04_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch08_s04_o01" numeration="arabic">
<li>Graph a hyperbola in standard form.</li>
<li>Determine the equation of a hyperbola given its graph.</li>
<li>Rewrite the equation of a hyperbola in standard form.</li>
<li>Identify a conic section given its equation.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch08_s04_s01" version="5.0" lang="en">
<h2 class="title editable block">The Hyperbola in Standard Form</h2>
<p class="para block" id="fwk-redden-ch08_s04_s01_p01">A <span class="margin_term"><a class="glossterm">hyperbola</a><span class="glossdef">The set of points in a plane whose distances from two fixed points, called foci, has an absolute difference that is equal to a positive constant.</span></span> is the set of points in a plane whose distances from two fixed points, called foci, has an absolute difference that is equal to a positive constant. In other words, if points <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0870" display="inline"><mrow><msub><mi>F</mi><mn>1</mn></msub></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0871" display="inline"><mrow><msub><mi>F</mi><mn>2</mn></msub></mrow></math></span> are the foci and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0872" display="inline"><mi>d</mi></math></span> is some given positive constant then <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0873" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span> is a point on the hyperbola if <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0874" display="inline"><mrow><mi>d</mi><mo>=</mo><mrow><mo>|</mo><mrow><msub><mi>d</mi><mn>1</mn></msub><mo>−</mo><msub><mi>d</mi><mn>2</mn></msub></mrow><mo>|</mo></mrow></mrow></math></span> as pictured below:</p>
<div class="informalfigure large block">
<img src="section_11/a1f2043cd1584f57925b89841098014b.png">
</div>
<p class="para editable block" id="fwk-redden-ch08_s04_s01_p03">In addition, a hyperbola is formed by the intersection of a cone with an oblique plane that intersects the base. It consists of two separate curves, called <span class="margin_term"><a class="glossterm">branches</a><span class="glossdef">The two separate curves of a hyperbola.</span></span>. Points on the separate branches of the graph where the distance is at a minimum are called <span class="margin_term"><a class="glossterm">vertices.</a><span class="glossdef">Points on the separate branches of a hyperbola where the distance is a minimum.</span></span> The midpoint between a hyperbola’s vertices is its center. Unlike a parabola, a hyperbola is asymptotic to certain lines drawn through the center. In this section, we will focus on graphing hyperbolas that open left and right or upward and downward.</p>
<div class="informalfigure large block">
<img src="section_11/0d8456e1f053f734cee5a43f831d1296.png">
</div>
<p class="para block" id="fwk-redden-ch08_s04_s01_p05">The asymptotes are drawn dashed as they are not part of the graph; they simply indicate the end behavior of the graph. The equation of a <span class="margin_term"><a class="glossterm">hyperbola opening left and right in standard form</a><span class="glossdef">The equation of a hyperbola written in the form <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0875" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span> The center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0876" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></math></span>, <em class="emphasis">a</em> defines the transverse axis, and <em class="emphasis">b</em> defines the conjugate axis.</span></span> follows:</p>
<p class="para block" id="fwk-redden-ch08_s04_s01_p06"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0877" display="block"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
<div class="informalfigure large block">
<img src="section_11/3333a997c65722c70df289bb1b7e13eb.png">
</div>
<p class="para block" id="fwk-redden-ch08_s04_s01_p08">Here the center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0878" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></math></span> and the vertices are <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0879" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>±</mo><mi>a</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> The equation of a <span class="margin_term"><a class="glossterm">hyperbola opening upward and downward in standard form</a><span class="glossdef">The equation of a hyperbola written in the form <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0880" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span> The center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0881" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></math></span>, <em class="emphasis">b</em> defines the transverse axis, and <em class="emphasis">a</em> defines the conjugate axis.</span></span> follows:</p>
<p class="para block" id="fwk-redden-ch08_s04_s01_p09"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0882" display="block"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
<div class="informalfigure large block">
<img src="section_11/a9dcd5884e392302b53231735d2c74a7.png">
</div>
<p class="para block" id="fwk-redden-ch08_s04_s01_p11">Here the center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0883" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></math></span> and the vertices are <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0884" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi><mo>±</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para block" id="fwk-redden-ch08_s04_s01_p12">The asymptotes are essential for determining the shape of any hyperbola. Given standard form, the asymptotes are lines passing through the center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0885" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></math></span> with slope <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0886" display="inline"><mrow><mi>m</mi><mo>=</mo><mo>±</mo><mfrac><mi>b</mi><mi>a</mi></mfrac></mrow><mo>.</mo></math></span> To easily sketch the asymptotes we make use of two special line segments through the center using <em class="emphasis">a</em> and <em class="emphasis">b</em>. Given any hyperbola, the <span class="margin_term"><a class="glossterm">transverse axis</a><span class="glossdef">The line segment formed by the vertices of a hyperbola.</span></span> is the line segment formed by its vertices. The <span class="margin_term"><a class="glossterm">conjugate axis</a><span class="glossdef">A line segment through the center of a hyperbola that is perpendicular to the transverse axis.</span></span> is the line segment through the center perpendicular to the transverse axis as pictured below:</p>
<div class="informalfigure large block">
<img src="section_11/c4b251ba8bbcb09f651d86f3a6c16a4d.png">
</div>
<p class="para block" id="fwk-redden-ch08_s04_s01_p14">The rectangle defined by the transverse and conjugate axes is called the <span class="margin_term"><a class="glossterm">fundamental rectangle</a><span class="glossdef">The rectangle formed using the endpoints of a hyperbolas, transverse and conjugate axes.</span></span>. The lines through the corners of this rectangle have slopes <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0887" display="inline"><mrow><mi>m</mi><mo>=</mo><mo>±</mo><mfrac><mi>b</mi><mi>a</mi></mfrac></mrow><mo>.</mo></math></span> These lines are the asymptotes that define the shape of the hyperbola. Therefore, given standard form, many of the properties of a hyperbola are apparent.</p>
<p class="para block" id="fwk-redden-ch08_s04_s01_p15"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para">Equation</p></th>
<th align="center"><p class="para">Center</p></th>
<th align="center"><p class="para"><em class="emphasis">a</em></p></th>
<th align="center"><p class="para"><em class="emphasis">b</em></p></th>
<th align="center"><p class="para">Opens</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0888" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>25</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>16</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0889" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0890" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>5</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0891" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>4</mn></mrow></math></span></p></td>
<td align="center"><p class="para">Left and right</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0892" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0893" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0894" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>3</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0895" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>6</mn></mrow></math></span></p></td>
<td align="center"><p class="para">Upward and downward</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0896" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>3</mn></mfrac><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0897" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0898" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0899" display="inline"><mrow><mi>b</mi><mo>=</mo><msqrt><mn>3</mn></msqrt></mrow></math></span></p></td>
<td align="center"><p class="para">Upward and downward</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0900" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>49</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>8</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0901" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0902" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>7</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0903" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></math></span></p></td>
<td align="center"><p class="para">Left and right</p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch08_s04_s01_p16">The graph of a hyperbola is completely determined by its center, vertices, and asymptotes.</p>
<div class="callout block" id="fwk-redden-ch08_s04_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch08_s04_s01_p17">Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0904" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s04_s01_p18">In this case, the expression involving <em class="emphasis">x</em> has a positive leading coefficient; therefore, the hyperbola opens left and right. Here <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0905" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mn>9</mn></msqrt><mo>=</mo><mn>3</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0906" display="inline"><mrow><mi>b</mi><mo>=</mo><msqrt><mn>4</mn></msqrt><mo>=</mo><mn>2</mn></mrow><mo>.</mo></math></span> From the center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0907" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>,</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span>, mark points 3 units left and right as well as 2 units up and down. Connect these points with a rectangle as follows:</p>
<div class="informalfigure large">
<img src="section_11/c6a0814f9a0f31fdacced13299687077.png">
</div>
<p class="para" id="fwk-redden-ch08_s04_s01_p20">The lines through the corners of this rectangle define the asymptotes.</p>
<div class="informalfigure large">
<img src="section_11/4ae171d1851f4c4d23f15f0a811b4529.png">
</div>
<p class="para" id="fwk-redden-ch08_s04_s01_p22">Use these dashed lines as a guide to graph the hyperbola opening left and right passing through the vertices.</p>
<p class="para" id="fwk-redden-ch08_s04_s01_p23">Answer:</p>
<div class="informalfigure large">
<img src="section_11/fe4cc56d6b89c8f71a4812e055774cdd.png">
</div>
</div>
<div class="callout block" id="fwk-redden-ch08_s04_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch08_s04_s01_p24">Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0908" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s04_s01_p25">In this case, the expression involving <em class="emphasis">y</em> has a positive leading coefficient; therefore, the hyperbola opens upward and downward. Here <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0909" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mrow><mn>36</mn></mrow></msqrt><mo>=</mo><mn>6</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0910" display="inline"><mrow><mi>b</mi><mo>=</mo><msqrt><mn>4</mn></msqrt><mo>=</mo><mn>2</mn></mrow><mo>.</mo></math></span> From the center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0911" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span> mark points 6 units left and right as well as 2 units up and down. Connect these points with a rectangle. The lines through the corners of this rectangle define the asymptotes.</p>
<div class="informalfigure large">
<img src="section_11/8d0fae2b7d2bf40948a42bf20f02197e.png">
</div>
<p class="para" id="fwk-redden-ch08_s04_s01_p27">Use these dashed lines as a guide to graph the hyperbola opening upward and downward passing through the vertices.</p>
<p class="para" id="fwk-redden-ch08_s04_s01_p28">Answer:</p>
<div class="informalfigure large">
<img src="section_11/b715577b958337d0b6f1f051ec20c0df.png">
</div>
</div>
<p class="para editable block" id="fwk-redden-ch08_s04_s01_p29"><strong class="emphasis bold">Note</strong>: When given a hyperbola opening upward and downward, as in the previous example, it is a common error to interchange the values for the center, <em class="emphasis">h</em> and <em class="emphasis">k</em>. This is the case because the quantity involving the variable <em class="emphasis">y</em> usually appears first in standard form. Take care to ensure that the <em class="emphasis">y</em>-value of the center comes from the quantity involving the variable <em class="emphasis">y</em> and that the <em class="emphasis">x</em>-value of the center is obtained from the quantity involving the variable <em class="emphasis">x</em>.</p>
<p class="para editable block" id="fwk-redden-ch08_s04_s01_p30">As with any graph, we are interested in finding the <em class="emphasis">x</em>- and <em class="emphasis">y</em>-intercepts.</p>
<div class="callout block" id="fwk-redden-ch08_s04_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch08_s04_s01_p31">Find the intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0912" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s04_s01_p32">To find the <em class="emphasis">x</em>-intercepts set <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0913" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>0</mn></mrow></math></span> and solve for <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch08_s04_s01_p33"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0914" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>0</mn></mstyle><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>1</mn><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>1</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch08_s04_s01_p34">Therefore there is only one <em class="emphasis">x</em>-intercept, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0915" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> To find the <em class="emphasis">y</em>-intercept set <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0916" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span> and solve for <em class="emphasis">y</em>.</p>
<p class="para" id="fwk-redden-ch08_s04_s01_p35"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0917" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>0</mn></mstyle><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>36</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>37</mn></mrow><mrow><mn>36</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>±</mo><mfrac><mrow><msqrt><mrow><mn>37</mn></mrow></msqrt></mrow><mn>6</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi><mo>−</mo><mn>2</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>±</mo><mfrac><mrow><msqrt><mrow><mn>37</mn></mrow></msqrt></mrow><mn>3</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mo>±</mo><mfrac><mrow><msqrt><mrow><mn>37</mn></mrow></msqrt></mrow><mn>3</mn></mfrac><mo>=</mo><mfrac><mrow><mn>6</mn><mo>±</mo><msqrt><mrow><mn>37</mn></mrow></msqrt></mrow><mn>3</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch08_s04_s01_p36">Therefore there are two <em class="emphasis">y</em>-intercepts, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0918" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mfrac><mrow><mn>6</mn><mo>−</mo><msqrt><mrow><mn>37</mn></mrow></msqrt></mrow><mn>3</mn></mfrac></mrow><mo>)</mo></mrow><mo>≈</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>0.03</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0919" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mfrac><mrow><mn>6</mn><mo>+</mo><msqrt><mrow><mn>37</mn></mrow></msqrt></mrow><mn>3</mn></mfrac></mrow><mo>)</mo></mrow><mo>≈</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>4.03</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> Take a moment to compare these to the sketch of the graph in the previous example.</p>
<p class="para" id="fwk-redden-ch08_s04_s01_p37">Answer: <em class="emphasis">x</em>-intercept: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0920" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>; <em class="emphasis">y</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0921" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mfrac><mrow><mn>6</mn><mo>−</mo><msqrt><mrow><mn>37</mn></mrow></msqrt></mrow><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0922" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mfrac><mrow><mn>6</mn><mo>+</mo><msqrt><mrow><mn>37</mn></mrow></msqrt></mrow><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch08_s04_s01_p38">Consider the hyperbola centered at the origin,</p>
<p class="para block" id="fwk-redden-ch08_s04_s01_p39"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0923" display="block"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>45</mn></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch08_s04_s01_p40">Standard form requires one side to be equal to 1. In this case, we can obtain standard form by dividing both sides by 45.</p>
<p class="para block" id="fwk-redden-ch08_s04_s01_p41"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0924" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><mrow><mn>45</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>45</mn></mrow><mrow><mn>45</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>45</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><mrow><mn>45</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>45</mn></mrow><mrow><mn>45</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>5</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mn>9</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch08_s04_s01_p42">This can be written as follows:</p>
<p class="para block" id="fwk-redden-ch08_s04_s01_p43"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0925" display="block"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>5</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch08_s04_s01_p44">In this form, it is clear that the center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0926" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0927" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mn>5</mn></msqrt></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0928" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>3</mn></mrow><mo>.</mo></math></span> The graph follows.</p>
<div class="informalfigure large block">
<img src="section_11/fd88fdbeb2860c3c78fc343efc8cac7a.png">
</div>
<div class="callout block" id="fwk-redden-ch08_s04_s01_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch08_s04_s01_p46"><strong class="emphasis bold">Try this!</strong> Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0929" display="inline"><mrow><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mrow><mn>25</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch08_s04_s01_p47">Answer:</p>
<div class="informalfigure large">
<img src="section_11/a211cf6a2197b559f297fd754568cdf2.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/Dc2Vw_IiT1Q" condition="http://img.youtube.com/vi/Dc2Vw_IiT1Q/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/Dc2Vw_IiT1Q" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch08_s04_s02" version="5.0" lang="en">
<h2 class="title editable block">The Hyperbola in General Form</h2>
<p class="para block" id="fwk-redden-ch08_s04_s02_p01">We have seen that the graph of a hyperbola is completely determined by its center, vertices, and asymptotes; which can be read from its equation in standard form. However, the equation is not always given in standard form. The equation of a <span class="margin_term"><a class="glossterm">hyperbola in general form</a><span class="glossdef">The equation of a hyperbola written in the form <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0930" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>q</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi></mrow></mtd><mtd columnalign="left"></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>+</mo><mi>d</mi><mi>y</mi><mo>+</mo><mi>e</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mrow></math></span> or <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0931" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>q</mi><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>c</mi><mi>x</mi></mrow></mtd><mtd columnalign="left"></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>+</mo><mi>d</mi><mi>y</mi><mo>+</mo><mi>e</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0932" display="inline"><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn>0</mn></mrow><mo>.</mo></math></span></span></span> follows:</p>
<p class="para block" id="fwk-redden-ch08_s04_s02_p02"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0933" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>q</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi><mi>y</mi><mo>+</mo><mi>e</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mtext> </mtext><mtext> </mtext><mrow><mstyle color="#007fbf"><mi>H</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>r</mi><mi>b</mi><mi>o</mi><mi>l</mi><mi>a</mi><mtext> </mtext><mi>o</mi><mi>p</mi><mi>e</mi><mi>n</mi><mi>s</mi><mtext> </mtext><mi>l</mi><mi>e</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>r</mi><mi>i</mi><mi>g</mi><mi>h</mi><mi>t</mi><mo>.</mo></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>q</mi><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi><mi>y</mi><mo>+</mo><mi>e</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mtext> </mtext><mtext> </mtext><mrow><mstyle color="#007fbf"><mi>H</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>r</mi><mi>b</mi><mi>o</mi><mi>l</mi><mi>a</mi><mtext> </mtext><mi>o</mi><mi>p</mi><mi>e</mi><mi>n</mi><mi>s</mi><mtext> </mtext><mi>u</mi><mi>p</mi><mi>w</mi><mi>a</mi><mi>r</mi><mi>d</mi><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>d</mi><mi>o</mi><mi>w</mi><mi>n</mi><mi>w</mi><mi>a</mi><mi>r</mi><mi>d</mi><mo>.</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span>
where <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0934" display="inline"><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn>0</mn></mrow><mo>.</mo></math></span> The steps for graphing a hyperbola given its equation in general form are outlined in the following example.</p>
<div class="callout block" id="fwk-redden-ch08_s04_s02_n01">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch08_s04_s02_p04">Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0935" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>32</mn><mi>x</mi><mo>−</mo><mn>54</mn><mi>y</mi><mo>−</mo><mn>53</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s04_s02_p05">Begin by rewriting the equation in standard form.</p>
<ul class="itemizedlist" id="fwk-redden-ch08_s04_s02_l01" mark="none">
<li>
<p class="para"><strong class="emphasis bold">Step 1:</strong> Group the terms with the same variables and move the constant to the right side. Factor so that the leading coefficient of each grouping is 1.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0936" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>32</mn><mi>x</mi><mo>−</mo><mn>54</mn><mi>y</mi><mo>−</mo><mn>53</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>32</mn><mi>x</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>54</mn><mi>y</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>53</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow><mo>−</mo><mn>9</mn><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>y</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>53</mn></mtd></mtr></mtable></math></span></p>
</li>
<li>
<p class="para"><strong class="emphasis bold">Step 2:</strong> Complete the square for each grouping. In this case, for the terms involving <em class="emphasis">x</em> use <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0937" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>8</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mn>4</mn><mn>2</mn></msup><mo>=</mo><mn>16</mn></mrow></math></span> and for the terms involving <em class="emphasis">y</em> use <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0938" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>6</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>9</mn></mrow><mo>.</mo></math></span> The factor in front of each grouping affects the value used to balance the equation on the right,</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0939" display="block"><mrow><mn>4</mn><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo></mstyle><mstyle color="#007fbf"><mn>16</mn></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>9</mn><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>y</mi><mstyle color="#007f3f"><mo>+</mo></mstyle><mstyle color="#007f3f"><mn>9</mn></mstyle></mrow><mo>)</mo></mrow><mo>=</mo><mn>53</mn><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo></mstyle><mstyle color="#007fbf"><mn>64</mn></mstyle><mstyle color="#007f3f"><mo>−</mo></mstyle><mstyle color="#007f3f"><mn>81</mn></mstyle></mrow></math></span></p>
<p class="para">Because of the distributive property, adding 16 inside of the first grouping is equivalent to adding <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0940" display="inline"><mrow><mn>4</mn><mo>⋅</mo><mn>16</mn><mo>=</mo><mn>64</mn></mrow><mo>.</mo></math></span> Similarly, adding 9 inside of the second grouping is equivalent to adding <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0941" display="inline"><mrow><mo>−</mo><mn>9</mn><mo>⋅</mo><mn>9</mn><mo>=</mo><mo>−</mo><mn>81</mn></mrow><mo>.</mo></math></span> Now factor and then divide to obtain 1 on the right side.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0942" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>4</mn><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>36</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mstyle color="#007fbf"><mn>36</mn></mstyle></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>36</mn></mrow><mrow><mstyle color="#007fbf"><mn>36</mn></mstyle></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>36</mn></mrow><mrow><mn>36</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></math></span></p>
</li>
<li>
<strong class="emphasis bold">Step 3:</strong> Determine the center, <em class="emphasis">a</em>, and <em class="emphasis">b</em>, and then use this information to sketch the graph. In this case, the center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0943" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0944" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mn>9</mn></msqrt><mo>=</mo><mn>3</mn></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0945" display="inline"><mrow><mi>b</mi><mo>=</mo><msqrt><mn>4</mn></msqrt><mo>=</mo><mn>2</mn></mrow><mo>.</mo></math></span> Because the leading coefficient of the expression involving <em class="emphasis">x</em> is positive and the coefficient of the expression involving <em class="emphasis">y</em> is negative, we graph a hyperbola opening left and right.</li>
</ul>
<p class="para" id="fwk-redden-ch08_s04_s02_p06">Answer:</p>
<div class="informalfigure large">
<img src="section_11/e93e9b5ecda5a980182e91cb99665a36.png">
</div>
</div>
<div class="callout block" id="fwk-redden-ch08_s04_s02_n01a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch08_s04_s02_p07"><strong class="emphasis bold">Try this!</strong> Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0946" display="inline"><mrow><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>40</mn><mi>y</mi><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>60</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch08_s04_s02_p08">Answer:</p>
<div class="informalfigure large">
<img src="section_11/9c50995638498673675a9e006417b62b.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/s4F6IZLgVhs" condition="http://img.youtube.com/vi/s4F6IZLgVhs/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/s4F6IZLgVhs" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch08_s04_s03" version="5.0" lang="en">
<h2 class="title editable block">Identifying the Conic Sections</h2>
<p class="para editable block" id="fwk-redden-ch08_s04_s03_p01">In this section, the challenge is to identify a conic section given its equation in general form. To distinguish between the conic sections, use the exponents and coefficients. If the equation is quadratic in only one variable and linear in the other, then its graph will be a parabola.</p>
<p class="para block" id="fwk-redden-ch08_s04_s03_p02"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="left"><p class="para"><strong class="emphasis bold">Parabola:</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0947" display="inline"><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></math></span></p></td>
<td align="center"><p class="para"> </p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0948" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mi>k</mi></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0949" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><msup><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mi>h</mi></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>c</mi></mtd></mtr></mtable></math></span></p></td>
</tr>
<tr>
<td align="center">
<p class="para"> </p>
<div class="informalfigure medium">
<img src="section_11/ad3a4667a3696d0376d19d00432ab6fa.png">
</div>
</td>
<td align="center">
<p class="para"> </p>
<div class="informalfigure medium">
<img src="section_11/3221bdacaef2aee47bc5fe4ac20e72bd.png">
</div>
</td>
</tr>
</tbody>
</table>
</div>
<p class="para block" id="fwk-redden-ch08_s04_s03_p03"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="left"><p class="para"><strong class="emphasis bold">Parabola:</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0950" display="inline"><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></math></span></p></td>
<td align="center"><p class="para"> </p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0951" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mi>k</mi></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0952" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><msup><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mi>h</mi></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>c</mi></mtd></mtr></mtable></math></span></p></td>
</tr>
<tr>
<td align="center">
<p class="para"> </p>
<div class="informalfigure medium">
<img src="section_11/4c3802d8ab9a2d6e448e73c42af3a1bc.png">
</div>
</td>
<td align="center">
<p class="para"> </p>
<div class="informalfigure medium">
<img src="section_11/f191058518db4fe375589dc136864cd6.png">
</div>
</td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch08_s04_s03_p04">If the equation is quadratic in both variables, where the coefficients of the squared terms are the same, then its graph will be a circle.</p>
<p class="para block" id="fwk-redden-ch08_s04_s03_p05"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="left"><p class="para">Circle:</p></th>
<th align="center"><p class="para"> </p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0954" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>r</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi><mi>y</mi><mo>+</mo><mi>e</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></math></span></p></td>
<td align="center">
<p class="para"> </p>
<div class="informalfigure medium">
<img src="section_11/3314b1767fe28199255a9a9612e5431f.png">
</div>
</td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch08_s04_s03_p06">If the equation is quadratic in both variables where the coefficients of the squared terms are different but have the same sign, then its graph will be an ellipse.</p>
<p class="para block" id="fwk-redden-ch08_s04_s03_p07"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="left"><p class="para"><strong class="emphasis bold">Ellipse:</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0955" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>></mo><mn>0</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0956" display="inline"><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn>0</mn></mrow></math></span></p></td>
<td align="center"><p class="para"> </p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0957" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi><mi>y</mi><mo>+</mo><mi>e</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></math></span></p></td>
<td align="center">
<p class="para"> </p>
<div class="informalfigure medium">
<img src="section_11/e075fbe9fa6903795bdf95eee2bc6662.png">
</div>
</td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch08_s04_s03_p08">If the equation is quadratic in both variables where the coefficients of the squared terms have different signs, then its graph will be a hyperbola.</p>
<p class="para block" id="fwk-redden-ch08_s04_s03_p09"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="left"><p class="para"><strong class="emphasis bold">Hyperbola:</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0958" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>></mo><mn>0</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0959" display="inline"><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn>0</mn></mrow></math></span></p></td>
<td align="center"><p class="para"> </p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0960" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>q</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi><mi>y</mi><mo>+</mo><mi>e</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0961" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>q</mi><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi><mi>y</mi><mo>+</mo><mi>e</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></math></span></p></td>
</tr>
<tr>
<td align="center">
<p class="para"> </p>
<div class="informalfigure medium">
<img src="section_11/e82a13fc4c06af299ebda321c506958e.png">
</div>
</td>
<td align="center">
<p class="para"> </p>
<div class="informalfigure medium">
<img src="section_11/b2845b8e4affef913136b83e874419e7.png">
</div>
</td>
</tr>
</tbody>
</table>
</div>
<div class="callout block" id="fwk-redden-ch08_s04_s03_n01">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch08_s04_s03_p10">Identify the graph of each equation as a parabola, circle, ellipse, or hyperbola.</p>
<ol class="orderedlist" id="fwk-redden-ch08_s04_s03_o01" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0962" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0963" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mo>=</mo><mn>0</mn></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0964" display="inline"><mrow><mi>x</mi><mo>−</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>y</mi><mo>+</mo><mn>11</mn><mo>=</mo><mn>0</mn></mrow></math></span></li>
</ol>
<p class="simpara">Solution:</p>
<ol class="orderedlist" id="fwk-redden-ch08_s04_s03_o02" numeration="loweralpha">
<li>
<p class="para">The equation is quadratic in both <em class="emphasis">x</em> and <em class="emphasis">y</em> where the leading coefficients for both variables is the same, 4.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0965" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para">This is an equation of a circle centered at the origin with radius 1/2.</p>
</li>
<li>
<p class="para">The equation is quadratic in both <em class="emphasis">x</em> and <em class="emphasis">y</em> where the leading coefficients for both variables have different signs.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0966" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><mrow><mn>12</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>12</mn></mrow><mrow><mn>12</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mn>6</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para">This is an equation of a hyperbola opening left and right centered at the origin.</p>
</li>
<li>
<p class="para">The equation is quadratic in <em class="emphasis">y</em> only.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0966a" display="block"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>y</mi><mo>−</mo><mn>11</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>y</mi><mo>+</mo><mtext></mtext><mtext></mtext><mtext></mtext><mtext></mtext><mtext></mtext><mtext></mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>+</mo><mn>11</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>y</mi><mstyle color="#007fbf"><mi> </mi><mo>+</mo><mn>9</mn></mstyle><mo stretchy="false">)</mo><mo>+</mo><mn>11</mn><mstyle color="#007fbf"><mi> </mi><mo>−</mo><mn>9</mn></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para">This is an equation of a parabola opening right with vertex <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0969" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
</li>
</ol>
<p class="para" id="fwk-redden-ch08_s04_s03_p11">Answer:</p>
<ol class="orderedlist" id="fwk-redden-ch08_s04_s03_o03" numeration="loweralpha">
<li>Circle</li>
<li>Hyperbola</li>
<li>Parabola</li>
</ol>
</div>
<div class="key_takeaways block" id="fwk-redden-ch08_s04_s03_n02">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch08_s04_s03_l01" mark="bullet">
<li>The graph of a hyperbola is completely determined by its center, vertices, and asymptotes.</li>
<li>The center, vertices, and asymptotes are apparent if the equation of a hyperbola is given in standard form: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0970" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span> or <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0971" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span>
</li>
<li>To graph a hyperbola, mark points <em class="emphasis">a</em> units left and right from the center and points <em class="emphasis">b</em> units up and down from the center. Use these points to draw the fundamental rectangle; the lines through the corners of this rectangle are the asymptotes. If the coefficient of <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0972" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> is positive, draw the branches of the hyperbola opening left and right through the points determined by <em class="emphasis">a</em>. If the coefficient of <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0973" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span> is positive, draw the branches of the hyperbola opening up and down through the points determined by <em class="emphasis">b</em>.</li>
<li>The orientation of the transverse axis depends the coefficient of <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0974" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0975" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span>
</li>
<li>If the equation of a hyperbola is given in general form <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0976" display="inline"><mrow><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>q</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi><mi>y</mi><mo>+</mo><mi>e</mi><mo>=</mo><mn>0</mn></mrow></math></span> or <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0977" display="inline"><mrow><mi>q</mi><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi><mi>y</mi><mo>+</mo><mi>e</mi><mo>=</mo><mn>0</mn></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0978" display="inline"><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn>0</mn></mrow></math></span>, group the terms with the same variables, and complete the square for both groupings to obtain standard form.</li>
<li>We recognize the equation of a hyperbola if it is quadratic in both <em class="emphasis">x</em> and <em class="emphasis">y</em> where the coefficients of the squared terms are opposite in sign.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch08_s04_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd01">
<h3 class="title">Part A: The Hyperbola in Standard Form</h3>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch08_s04_qs01_p01"><strong class="emphasis bold">Given the equation of a hyperbola in standard form, determine its center, which way the graph opens, and the vertices.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0979" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>16</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0985" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>25</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>64</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0991" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>5</mn></mfrac><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0997" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>12</mn></mrow></mfrac><mo>−</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1003" display="inline"><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>25</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>100</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1009" display="inline"><mrow><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>5</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>45</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd01_qd02" start="7">
<p class="para" id="fwk-redden-ch08_s04_qs01_p14"><strong class="emphasis bold">Determine the standard form for the equation of a hyperbola given the following information.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p15">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1015" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1016" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>6</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1017" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>3</mn></mrow></math></span>, opens left and right.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p17">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1019" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>9</mn><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1020" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>7</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1021" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>2</mn></mrow></math></span>, opens up and down.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p19">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1023" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>10</mn><mo>,</mo><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1024" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mn>7</mn></msqrt></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1025" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>5</mn><msqrt><mn>2</mn></msqrt></mrow></math></span>, opens up and down.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p21">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1027" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>7</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1028" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1029" display="inline"><mrow><mi>b</mi><mo>=</mo><msqrt><mn>5</mn></msqrt></mrow></math></span>, opens left and right.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p23">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1031" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1032" display="inline"><mrow><mi>a</mi><mo>=</mo><msqrt><mn>2</mn></msqrt></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1033" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>1</mn></mrow></math></span>,opens up and down.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p25">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1035" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1036" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>2</mn><msqrt><mn>6</mn></msqrt></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1037" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>4</mn></mrow></math></span>, opens left and right.</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd01_qd03" start="13">
<p class="para" id="fwk-redden-ch08_s04_qs01_p27"><strong class="emphasis bold">Graph.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p28"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1039" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>16</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p30"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1040" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>25</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p32"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1041" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>16</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>1</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p34"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1042" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p36"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1043" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>16</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p38"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1044" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p40"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1045" display="inline"><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>36</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p42"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1046" display="inline"><mrow><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>64</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p44"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1047" display="inline"><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>25</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>100</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p46"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1048" display="inline"><mrow><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>144</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p48"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1049" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>12</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p50"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1050" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>8</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p52"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1051" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>5</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p54"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1052" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>3</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>18</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p56"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1053" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p58"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1054" display="inline"><mrow><mn>7</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>14</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p60"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1055" display="inline"><mrow><mn>6</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p62"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1056" display="inline"><mrow><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>30</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd01_qd04" start="31">
<p class="para" id="fwk-redden-ch08_s04_qs01_p64"><strong class="emphasis bold">Find the <em class="emphasis">x</em>- and <em class="emphasis">y</em>-intercepts.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p65"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1057" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p67"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1059" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>16</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p69"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1062" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p71"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1064" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>16</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p73"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1067" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1069" display="inline"><mrow><mn>6</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p77"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1072" display="inline"><mrow><mn>36</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p79"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1074" display="inline"><mrow><mn>6</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p81">Find the equation of the hyperbola with vertices <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1076" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>±</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span> and a conjugate axis that measures 12 units.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p83">Find the equation of the hyperbola with vertices <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1078" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>4</mn><mo>,</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1079" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>4</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span> and a conjugate axis that measures 6 units.</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd02">
<h3 class="title">Part B: The Hyperbola in General Form</h3>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd02_qd01" start="41">
<p class="para" id="fwk-redden-ch08_s04_qs01_p85"><strong class="emphasis bold">Rewrite in standard form and graph.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p86"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1081" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mi>x</mi><mo>+</mo><mn>54</mn><mi>y</mi><mo>−</mo><mn>101</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p88"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1083" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>18</mn><mi>x</mi><mo>−</mo><mn>100</mn><mi>y</mi><mo>−</mo><mn>316</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p90"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1085" display="inline"><mrow><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>64</mn><mi>x</mi><mo>+</mo><mn>8</mn><mi>y</mi><mo>−</mo><mn>124</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p92"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1087" display="inline"><mrow><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>24</mn><mi>x</mi><mo>−</mo><mn>72</mn><mi>y</mi><mo>+</mo><mn>72</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p94"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1089" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>36</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>72</mn><mi>x</mi><mo>−</mo><mn>12</mn><mi>y</mi><mo>−</mo><mn>36</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p97"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1091" display="inline"><mrow><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>36</mn><mi>y</mi><mo>+</mo><mn>11</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p99"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1093" display="inline"><mrow><mn>36</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>24</mn><mi>y</mi><mo>−</mo><mn>180</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p101"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1095" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>24</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p103"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1097" display="inline"><mrow><mn>25</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>64</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>200</mn><mi>x</mi><mo>+</mo><mn>640</mn><mi>y</mi><mo>−</mo><mn>2,800</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p105"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1099" display="inline"><mrow><mn>49</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>40</mn><mi>x</mi><mo>+</mo><mn>490</mn><mi>y</mi><mo>+</mo><mn>929</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p107"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1101" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>24</mn><mi>x</mi><mo>+</mo><mn>8</mn><mi>y</mi><mo>+</mo><mn>34</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p109"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1103" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>24</mn><mi>x</mi><mo>+</mo><mn>80</mn><mi>y</mi><mo>−</mo><mn>196</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p111"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1105" display="inline"><mrow><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi><mo>−</mo><mn>16</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p113"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1107" display="inline"><mrow><mn>12</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>40</mn><mi>x</mi><mo>+</mo><mn>48</mn><mi>y</mi><mo>−</mo><mn>92</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p115"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1109" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>16</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>16</mn><mi>y</mi><mo>−</mo><mn>11</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p117"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1111" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>16</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p119"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1113" display="inline"><mrow><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>36</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>108</mn><mi>x</mi><mo>−</mo><mn>117</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p121"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1115" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mo>−</mo><mn>33</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd02_qd02" start="59">
<p class="para" id="fwk-redden-ch08_s04_qs01_p123"><strong class="emphasis bold">Given the general form, determine the intercepts.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p124"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1117" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>11</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>−</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p126"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1121" display="inline"><mrow><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>9</mn><mi>y</mi><mo>−</mo><mn>9</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p128"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1124" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p130"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1126" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>y</mi><mo>−</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>16</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p132"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1130" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p134"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1135" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd02_qd03" start="65">
<p class="para" id="fwk-redden-ch08_s04_qs01_p136"><strong class="emphasis bold">Find the equations of the asymptotes to the given hyperbola.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p137"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1138" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>16</mn></mrow></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p139"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1141" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p141"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1144" display="inline"><mrow><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>24</mn><mi>y</mi><mo>−</mo><mn>96</mn><mi>x</mi><mo>+</mo><mn>44</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p143"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1147" display="inline"><mrow><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>y</mi><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd02_qd04" start="69">
<p class="para" id="fwk-redden-ch08_s04_qs01_p145"><strong class="emphasis bold">Given the graph of a hyperbola, determine its equation in general form.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa69">
<div class="question">
<div class="informalfigure large">
<img src="section_11/36dccf1e26f02f048fbd1d877667a147.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa70">
<div class="question">
<div class="informalfigure large">
<img src="section_11/34f2338ad74b7c7d6f1a21b6c8c4d186.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa71">
<div class="question">
<div class="informalfigure large">
<img src="section_11/a7bc4eadfff45a64281536baae8dc741.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa72">
<div class="question">
<div class="informalfigure large">
<img src="section_11/5a4c64dcf71ab2c3f0a80ed545fad3e4.png">
</div>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd03">
<h3 class="title">Part C: Identifying the Conic Sections</h3>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd03_qd01" start="73">
<p class="para" id="fwk-redden-ch08_s04_qs01_p154"><strong class="emphasis bold">Identify the following as the equation of a line, parabola, circle, ellipse, or hyperbola.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p155"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1154" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>23</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p157"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1155" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p159"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1156" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>14</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p161"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1157" display="inline"><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>24</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p163"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1158" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>36</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p165"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1159" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>32</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p167"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1160" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p169"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1161" display="inline"><mrow><mi>x</mi><mo>−</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p171"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1162" display="inline"><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p173"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1163" display="inline"><mrow><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>144</mn><mi>x</mi><mo>−</mo><mn>12</mn><mi>y</mi><mo>+</mo><mn>641</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd03_qd02" start="83">
<p class="para" id="fwk-redden-ch08_s04_qs01_p175"><strong class="emphasis bold">Identify the conic sections and rewrite in standard form.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p176"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1164" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>y</mi><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>11</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p178"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1166" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi><mo>+</mo><mn>44</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p180"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1168" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>12</mn><mi>y</mi><mo>−</mo><mn>18</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p182"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1170" display="inline"><mrow><mn>25</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>36</mn><mi>x</mi><mo>−</mo><mn>50</mn><mi>y</mi><mo>−</mo><mn>187</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p184"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1172" display="inline"><mrow><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>84</mn><mi>x</mi><mo>+</mo><mn>16</mn><mi>y</mi><mo>+</mo><mn>240</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p186"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1174" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>80</mn><mi>x</mi><mo>+</mo><mn>399</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p188"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1176" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>32</mn><mi>y</mi><mo>+</mo><mn>29</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p190"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1178" display="inline"><mrow><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>32</mn><mi>x</mi><mo>+</mo><mn>20</mn><mi>y</mi><mo>−</mo><mn>25</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p192"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1180" display="inline"><mrow><mn>9</mn><mi>x</mi><mo>−</mo><mn>18</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>y</mi><mo>+</mo><mn>7</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa92">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p194"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1182" display="inline"><mrow><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>24</mn><mi>x</mi><mo>−</mo><mn>48</mn><mi>y</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd04">
<h3 class="title">Part D: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch08_s04_qs01_qd04_qd01" start="93">
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa93">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p196">Develop a formula for the equations of the asymptotes of a hyperbola. Share it along with an example on the discussion board.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa94">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p197">Make up your own equation of a hyperbola, write it in general form, and graph it.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa95">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p198">Do all hyperbolas have intercepts? What are the possible numbers of intercepts for a hyperbola? Explain.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa96">
<div class="question">
<p class="para" id="fwk-redden-ch08_s04_qs01_p199">Research and discuss real-world examples of hyperbolas.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch08_s04_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s04_qs01_p03_ans">Center: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0980" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>6</mn><mo>,</mo><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0981" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>4</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0982" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>3</mn></mrow></math></span>; opens left and right; vertices: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0983" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0984" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>10</mn><mo>,</mo><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s04_qs01_p07_ans">Center: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0992" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0993" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0994" display="inline"><mrow><mi>b</mi><mo>=</mo><msqrt><mn>5</mn></msqrt></mrow></math></span>; opens upward and downward; vertices: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0995" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>9</mn><mo>−</mo><msqrt><mn>5</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0996" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>9</mn><mo>+</mo><msqrt><mn>5</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s04_qs01_p11_ans">Center: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1004" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1005" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>2</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1006" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>5</mn></mrow></math></span>; opens upward and downward; vertices: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1007" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>15</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m1008" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s04_qs01_p16_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1018" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>36</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa08_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s04_qs01_p20_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1026" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>50</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>7</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa10_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s04_qs01_p24_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m1034" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mn>1</mn></mfrac><mo>−</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa12_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa13_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/7513b81b369f98ac567e5f6085bdc005.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa14_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s04_qs01_qa15_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/7d68d36f6b7befbbc9a256b05d73c808.png">