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<!DOCTYPE html>
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<title>Circles</title>
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<div id="book-content">
<div class="section" id="fwk-redden-ch08_s02" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">8.2</span> Circles</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch08_s02_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch08_s02_o01" numeration="arabic">
<li>Graph a circle in standard form.</li>
<li>Determine the equation of a circle given its graph.</li>
<li>Rewrite the equation of a circle in standard form.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch08_s02_s01" version="5.0" lang="en">
<h2 class="title editable block">The Circle in Standard Form</h2>
<p class="para editable block" id="fwk-redden-ch08_s02_s01_p01">A <span class="margin_term"><a class="glossterm">circle</a><span class="glossdef">A circle is the set of points in a plane that lie a fixed distance from a given point, called the center.</span></span> is the set of points in a plane that lie a fixed distance, called the <span class="margin_term"><a class="glossterm">radius</a><span class="glossdef">The fixed distance from the center of a circle to any point on the circle.</span></span>, from any point, called the center. The <span class="margin_term"><a class="glossterm">diameter</a><span class="glossdef">The length of a line segment passing through the center of a circle whose endpoints are on the circle.</span></span> is the length of a line segment passing through the center whose endpoints are on the circle. In addition, a circle can be formed by the intersection of a cone and a plane that is perpendicular to the axis of the cone:</p>
<div class="informalfigure large block">
<img src="section_11/6666984a93265d78368a7db32a0e4435.png">
</div>
<p class="para block" id="fwk-redden-ch08_s02_s01_p03">In a rectangular coordinate plane, where the center of a circle with radius <em class="emphasis">r</em> is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0351" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></math></span>, we have</p>
<div class="informalfigure large block">
<img src="section_11/042678cf13cc2235520c64347fedd75c.png">
</div>
<p class="para block" id="fwk-redden-ch08_s02_s01_p05">Calculate the distance between <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0352" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0353" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span> using the distance formula,</p>
<p class="para block" id="fwk-redden-ch08_s02_s01_p06"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0354" display="block"><mrow><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt><mo>=</mo><mi>r</mi></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch08_s02_s01_p07">Squaring both sides leads us to the equation of a <span class="margin_term"><a class="glossterm">circle in standard form</a><span class="glossdef">The equation of a circle written in the form <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0355" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0356" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></math></span> is the center and <em class="emphasis">r</em> is the radius.</span></span>,</p>
<p class="para block" id="fwk-redden-ch08_s02_s01_p08"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0357" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch08_s02_s01_p09">In this form, the center and radius are apparent. For example, given the equation <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0358" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mtext> </mtext><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>16</mn></mrow></math></span> we have,</p>
<p class="para block" id="fwk-redden-ch08_s02_s01_p10"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0359" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><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></mtd><mtd><mo>+</mo></mtd><mtd><mo> </mo><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mtd></mtr><mtr><mtd><mspace width="1.0em"></mspace><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd><mrow></mrow></mtd><mtd><mspace width="1.5em"></mspace><mstyle color="#007fbf"><mo>↓</mo></mstyle></mtd><mtd><mrow></mrow></mtd><mtd><mstyle color="#007fbf"><mo>↓</mo></mstyle><mspace width="1.5em"></mspace></mtd></mtr><mtr><mtd><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mstyle color="#007fbf"><mn>2</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>+</mo></mtd><mtd><mrow><msup><mrow><mrow><mo>[</mo><mrow><mi>y</mi><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>5</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mstyle color="#007fbf"><mn>4</mn></mstyle><mn>2</mn></msup></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch08_s02_s01_p11">In this case, the center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0360" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0361" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>4</mn></mrow><mo>.</mo></math></span> More examples follow:</p>
<p class="para block" id="fwk-redden-ch08_s02_s01_p12"></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">Radius</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0362" display="inline"><mrow><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><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>25</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0363" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><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_m0364" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>5</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0365" display="inline"><mrow><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><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><mn>7</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0366" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</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_m0367" display="inline"><mrow><mi>r</mi><mo>=</mo><msqrt><mn>7</mn></msqrt></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0368" display="inline"><mrow><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><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>1</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0369" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mn>3</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_m0370" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0371" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>8</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0372" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>6</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_m0373" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch08_s02_s01_p13">The graph of a circle is completely determined by its center and radius.</p>
<div class="callout block" id="fwk-redden-ch08_s02_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch08_s02_s01_p14">Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0374" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>16</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s02_s01_p15">Written in this form we can see that the center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0375" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span> and that the radius <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0376" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>4</mn></mrow></math></span> units. From the center mark points 4 units up and down as well as 4 units left and right.</p>
<div class="informalfigure large">
<img src="section_11/4a6d4985550856b71cda468ba5771f41.png">
</div>
<p class="para" id="fwk-redden-ch08_s02_s01_p17">Then draw in the circle through these four points.</p>
<p class="para" id="fwk-redden-ch08_s02_s01_p18">Answer:</p>
<div class="informalfigure large">
<img src="section_11/298ead43e24c401b0472c9671b64ea57.png">
</div>
</div>
<p class="para editable block" id="fwk-redden-ch08_s02_s01_p19">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_s02_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch08_s02_s01_p20">Find the intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0377" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>16</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s02_s01_p21">To find the <em class="emphasis">y</em>-intercepts set <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0378" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span>:</p>
<p class="para" id="fwk-redden-ch08_s02_s01_p22"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0379" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>16</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><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><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>16</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>4</mn><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>16</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch08_s02_s01_p23">For this equation, we can solve by extracting square roots.</p>
<p class="para" id="fwk-redden-ch08_s02_s01_p24"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0380" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><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></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>12</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi>y</mi><mo>+</mo><mn>5</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mn>12</mn></mrow></msqrt></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi>y</mi><mo>+</mo><mn>5</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>5</mn><mo>±</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch08_s02_s01_p25">Therefore, the <em class="emphasis">y</em>-intercepts are <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0381" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>5</mn><mo>−</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0382" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>5</mn><mo>+</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> To find the <em class="emphasis">x</em>-intercepts set <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0383" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>0</mn></mrow></math></span>:</p>
<p class="para" id="fwk-redden-ch08_s02_s01_p26"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0384" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>16</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>0</mn></mstyle><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>16</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>25</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>16</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><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></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>9</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mo>−</mo><mn>9</mn></mrow></msqrt></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mo>±</mo><mn>3</mn><mi>i</mi></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch08_s02_s01_p27">And because the solutions are complex we conclude that there are no real <em class="emphasis">x</em>-intercepts. Note that this does make sense given the graph.</p>
<div class="informalfigure large">
<img src="section_11/ea7307d4ab2655bac106c9d9cb2cce01.png">
</div>
<p class="para" id="fwk-redden-ch08_s02_s01_p29">Answer: <em class="emphasis">x</em>-intercepts: none; <em class="emphasis">y</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0385" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>5</mn><mo>−</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0386" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>5</mn><mo>+</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch08_s02_s01_p30">Given the center and radius of a circle, we can find its equation.</p>
<div class="callout block" id="fwk-redden-ch08_s02_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch08_s02_s01_p31">Graph the circle with radius <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0387" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>3</mn></mrow></math></span> units centered at <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0388" 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> Give its equation in standard form and determine the intercepts.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s02_s01_p32">Given that the center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0389" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span> and the radius is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0390" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>3</mn></mrow></math></span> we sketch the graph as follows:</p>
<div class="informalfigure large">
<img src="section_11/d3971576f235c8ddb75b91876576735d.png">
</div>
<p class="para" id="fwk-redden-ch08_s02_s01_p34">Substitute <em class="emphasis">h</em>, <em class="emphasis">k</em>, and <em class="emphasis">r</em> to find the equation in standard form. Since <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0391" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0392" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>3</mn></mrow></math></span> we have,</p>
<p class="para" id="fwk-redden-ch08_s02_s01_p35"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0393" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>[</mo><mrow><mi>x</mi><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo></mstyle><mstyle color="#007fbf"><mn>1</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mstyle color="#007fbf"><mn>0</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mstyle color="#007fbf"><mn>3</mn></mstyle><mn>2</mn></msup></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><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><msup><mi>y</mi><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch08_s02_s01_p36">The equation of the circle is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0394" display="inline"><mrow><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><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>9</mn></mrow></math></span>, use this to determine the <em class="emphasis">y</em>-intercepts.</p>
<p class="para" id="fwk-redden-ch08_s02_s01_p37"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0395" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><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><msup><mi>y</mi><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd><mtd><mtext> </mtext><mtext> </mtext><mrow><mstyle color="#007fbf"><mi>S</mi><mi>e</mi><mi>t</mi><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0</mn><mtext> </mtext><mi>t</mi><mi>o</mi><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>s</mi><mi>o</mi><mi>l</mi><mi>v</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>y</mi><mo>.</mo></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><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><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>1</mn><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mn>8</mn></msqrt></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></mtd><mtd><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch08_s02_s01_p38">Therefore, the <em class="emphasis">y</em>-intercepts are <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0396" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0397" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> To find the <em class="emphasis">x</em>-intercepts algebraically, set <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0398" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>0</mn></mrow></math></span> and solve for <em class="emphasis">x</em>; this is left for the reader as an exercise.</p>
<div class="informalfigure large">
<img src="section_11/7bc3fe416f7aa6a938a7f5425d8ea80a.png">
</div>
<p class="para" id="fwk-redden-ch08_s02_s01_p40">Answer: Equation: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0399" display="inline"><mrow><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><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>9</mn></mrow></math></span>; <em class="emphasis">y</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0400" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0401" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span>; <em class="emphasis">x</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0402" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0403" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch08_s02_s01_p41">Of particular importance is the <span class="margin_term"><a class="glossterm">unit circle</a><span class="glossdef">The circle centered at the origin with radius 1; its equation is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0404" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></span></span>,</p>
<p class="para block" id="fwk-redden-ch08_s02_s01_p42"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0405" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch08_s02_s01_p43">Or,</p>
<p class="para block" id="fwk-redden-ch08_s02_s01_p44"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0406" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mn>1</mn><mn>2</mn></msup></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch08_s02_s01_p45">In this form, it should be clear that the center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0407" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span> and that the radius is 1 unit. Furthermore, if we solve for <em class="emphasis">y</em> we obtain two functions:</p>
<p class="para block" id="fwk-redden-ch08_s02_s01_p46"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0408" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch08_s02_s01_p47">The function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0409" display="inline"><mrow><mi>y</mi><mo>=</mo><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow></math></span> is the top half of the circle and the function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0410" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow></math></span> is the bottom half of the unit circle:</p>
<div class="informalfigure large block">
<img src="section_11/3942b6a7877027b67716e7d43e36aa00.png">
</div>
<div class="callout block" id="fwk-redden-ch08_s02_s01_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch08_s02_s01_p49"><strong class="emphasis bold">Try this!</strong> Graph and label the intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0411" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><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><mn>25</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch08_s02_s01_p50">Answer:</p>
<div class="informalfigure large">
<img src="section_11/a856a74b448b5e255d1d955a925b81bf.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/KeKt9k6IDCk" condition="http://img.youtube.com/vi/KeKt9k6IDCk/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/KeKt9k6IDCk" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch08_s02_s02" version="5.0" lang="en">
<h2 class="title editable block">The Circle in General Form</h2>
<p class="para block" id="fwk-redden-ch08_s02_s02_p01">We have seen that the graph of a circle is completely determined by the center and radius 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">circle in general form</a><span class="glossdef">The equation of a circle written in the form <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0412" display="inline"><mrow><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><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></span></span> follows:</p>
<p class="para block" id="fwk-redden-ch08_s02_s02_p02"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0413" display="block"><mrow><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><mo>=</mo><mn>0</mn></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch08_s02_s02_p03">Here <em class="emphasis">c</em>, <em class="emphasis">d</em>, and <em class="emphasis">e</em> are real numbers. The steps for graphing a circle given its equation in general form follow.</p>
<div class="callout block" id="fwk-redden-ch08_s02_s02_n01">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch08_s02_s02_p04">Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0414" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>+</mo><mn>13</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s02_s02_p05">Begin by rewriting the equation in standard form.</p>
<ul class="itemizedlist" id="fwk-redden-ch08_s02_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. In this case, subtract 13 on both sides and group the terms involving <em class="emphasis">x</em> and the terms involving <em class="emphasis">y</em> as follows.</p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0415" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>+</mo><mn>13</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>y</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>13</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
</li>
<li>
<p class="para"><strong class="emphasis bold">Step 2:</strong> Complete the square for each grouping. The idea is to add the value that completes the square, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0416" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span>, to both sides for both groupings, and then factor. For the terms involving <em class="emphasis">x</em> use <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0417" 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><mn>3</mn><mn>2</mn></msup><mo>=</mo><mn>9</mn></mrow></math></span> and for the terms involving <em class="emphasis">y</em> use <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0418" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>8</mn></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>16</mn></mrow><mo>.</mo></math></span></p>
<p class="para"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0419" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo></mstyle><mstyle color="#007fbf"><mn>9</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>y</mi><mstyle color="#007f3f"><mo>+</mo></mstyle><mstyle color="#007f3f"><mn>16</mn></mstyle></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>13</mn><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo></mstyle><mstyle color="#007fbf"><mn>9</mn></mstyle><mstyle color="#007f3f"><mo>+</mo></mstyle><mstyle color="#007f3f"><mn>16</mn></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><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><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>12</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
</li>
<li>
<strong class="emphasis bold">Step 3:</strong> Determine the center and radius from the equation in standard form. In this case, the center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0420" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span> and the radius <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0421" display="inline"><mrow><mi>r</mi><mo>=</mo><msqrt><mrow><mn>12</mn></mrow></msqrt><mo>=</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mo>.</mo></math></span>
</li>
<li>
<strong class="emphasis bold">Step 4:</strong> From the center, mark the radius vertically and horizontally and then sketch the circle through these points.</li>
</ul>
<div class="informalfigure large">
<img src="section_11/1158d0423f6d2e2295ff4b155fbc9f06.png">
</div>
<p class="para" id="fwk-redden-ch08_s02_s02_p07">Answer:</p>
<div class="informalfigure large">
<img src="section_11/95fa940266f1558fbd6b86ae2aaa58fe.png">
</div>
</div>
<div class="callout block" id="fwk-redden-ch08_s02_s02_n02">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch08_s02_s02_p08">Determine the center and radius: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0422" 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>8</mn><mi>x</mi><mo>+</mo><mn>12</mn><mi>y</mi><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch08_s02_s02_p09">We can obtain the general form by first dividing both sides by 4.</p>
<p class="para" id="fwk-redden-ch08_s02_s02_p10"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0423" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mfrac><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>8</mn><mi>x</mi><mo>+</mo><mn>12</mn><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mn>4</mn></mfrac></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>0</mn><mn>4</mn></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="right"><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>3</mn><mi>y</mi><mo>−</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch08_s02_s02_p11">Now that we have the general form for a circle, where both terms of degree two have a leading coefficient of 1, we can use the steps for rewriting it in standard form. Begin by adding <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0424" display="inline"><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math></span> to both sides and group variables that are the same.</p>
<p class="para" id="fwk-redden-ch08_s02_s02_p12"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0425" display="block"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>y</mi><mo>+</mo><mo>___</mo></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch08_s02_s02_p13">Next complete the square for both groupings. Use <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0426" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>2</mn></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span> for the first grouping and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0427" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mfrac><mn>9</mn><mn>4</mn></mfrac></mrow></math></span> for the second grouping.</p>
<p class="para" id="fwk-redden-ch08_s02_s02_p14"><span class="informalequation"><math xml:id="fwk-redden-ch08_m0428" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo></mstyle><mstyle color="#007fbf"><mn>1</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>y</mi><mstyle color="#007f3f"><mo>+</mo></mstyle><mstyle color="#007f3f"><mfrac><mn>9</mn><mn>4</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo></mstyle><mstyle color="#007fbf"><mn>1</mn></mstyle><mstyle color="#007f3f"><mo>+</mo></mstyle><mstyle color="#007f3f"><mfrac><mn>9</mn><mn>4</mn></mfrac></mstyle></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><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><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>16</mn></mrow><mn>4</mn></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><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><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch08_s02_s02_p15">Answer: Center: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0429" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>; radius: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0430" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch08_s02_s02_p16">In summary, to convert from standard form to general form we multiply, and to convert from general form to standard form we complete the square.</p>
<div class="informalfigure large block">
<img src="section_11/be7fda011544ac78a5e893656a9eef8d.png">
</div>
<div class="callout block" id="fwk-redden-ch08_s02_s02_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch08_s02_s02_p18"><strong class="emphasis bold">Try this!</strong> Graph: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0431" 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>21</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch08_s02_s02_p19">Answer:</p>
<div class="informalfigure large">
<img src="section_11/6a4aa4b08a41eabd97b7c9d1a44e38eb.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/Ms8NESnqs6s" condition="http://img.youtube.com/vi/Ms8NESnqs6s/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/Ms8NESnqs6s" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<div class="key_takeaways block" id="fwk-redden-ch08_s02_s02_n03">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch08_s02_s02_l02" mark="bullet">
<li>The graph of a circle is completely determined by its center and radius.</li>
<li>Standard form for the equation of a circle is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0432" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span> The center is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0433" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow></math></span> and the radius measures <em class="emphasis">r</em> units.</li>
<li>To graph a circle mark points <em class="emphasis">r</em> units up, down, left, and right from the center. Draw a circle through these four points.</li>
<li>If the equation of a circle is given in general form <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0434" display="inline"><mrow><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><mo>=</mo><mn>0</mn></mrow></math></span>, group the terms with the same variables, and complete the square for both groupings. This will result in standard form, from which we can read the circle’s center and radius.</li>
<li>We recognize the equation of a circle if it is quadratic in both <em class="emphasis">x</em> and <em class="emphasis">y</em> where the coefficient of the squared terms are the same.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch08_s02_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch08_s02_qs01_qd01">
<h3 class="title">Part A: The Circle in Standard Form</h3>
<ol class="qandadiv" id="fwk-redden-ch08_s02_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch08_s02_qs01_p01"><strong class="emphasis bold">Determine the center and radius given the equation of a circle in standard form.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0435" display="inline"><mrow><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><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>4</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_s02_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0438" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>121</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0441" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0444" display="inline"><mrow><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><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_s02_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0447" display="inline"><mrow><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><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>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0450" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s02_qs01_qd01_qd02" start="7">
<p class="para" id="fwk-redden-ch08_s02_qs01_p14"><strong class="emphasis bold">Determine the standard form for the equation of the circle given its center and radius.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p15">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0453" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>,</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></math></span> with radius <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0454" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>7</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p17">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0456" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></math></span> with radius <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0457" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>5</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p19">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0459" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>6</mn><mo>,</mo><mo>−</mo><mn>11</mn></mrow><mo>)</mo></mrow></mrow></math></span> with radius <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0460" display="inline"><mrow><mi>r</mi><mo>=</mo><msqrt><mn>2</mn></msqrt></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p21">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0462" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span> with radius <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0463" display="inline"><mrow><mi>r</mi><mo>=</mo><msqrt><mn>6</mn></msqrt></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p23">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0465" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span> with radius <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0466" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>2</mn><msqrt><mn>5</mn></msqrt></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p25">Center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0468" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span> with radius <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0469" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>3</mn><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow><mo>.</mo></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s02_qs01_qd01_qd03" start="13">
<p class="para" id="fwk-redden-ch08_s02_qs01_p27"><strong class="emphasis bold">Graph.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p28"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0471" display="inline"><mrow><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><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><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p30"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0472" display="inline"><mrow><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><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>25</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p32"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0473" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p34"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0474" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>4</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_s02_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p36"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0475" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p38"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0476" display="inline"><mrow><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><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p40"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0477" 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></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p42"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0478" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p44"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0479" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p46"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0480" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p48"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0481" display="inline"><mrow><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><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>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p50"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0482" display="inline"><mrow><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><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><mn>15</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s02_qs01_qd01_qd04" start="25">
<p class="para" id="fwk-redden-ch08_s02_qs01_p52"><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_s02_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p53"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0483" display="inline"><mrow><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><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><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p55"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0486" display="inline"><mrow><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><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>25</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p57"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0490" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p59"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0493" display="inline"><mrow><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><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p61"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0496" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>50</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p63"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0499" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>20</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p65"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0501" display="inline"><mrow><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><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p67"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0502" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>20</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>400</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s02_qs01_qd01_qd05" start="33">
<p class="para" id="fwk-redden-ch08_s02_qs01_p69"><strong class="emphasis bold">Find the equation of the circle.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p70">Circle with center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0505" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span> passing through <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0506" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p72">Circle with center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0508" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span> passing through <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0509" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p74">Circle whose diameter is defined by <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0511" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0512" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p76">Circle whose diameter is defined by <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0514" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn><mo>,</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0515" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p78">Circle with center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0517" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span> and area <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0518" display="inline"><mrow><mn>9</mn><mi>π</mi></mrow></math></span> square units.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p80">Circle with center <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0520" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>8</mn><mo>,</mo><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span> and circumference <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0521" display="inline"><mrow><mn>12</mn><mi>π</mi></mrow></math></span> square units.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p82">Find the area of the circle with equation <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0523" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><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>7</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p84">Find the circumference of the circle with equation <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0525" display="inline"><mrow><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><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>8</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s02_qs01_qd02">
<h3 class="title">Part B: The Circle in General Form</h3>
<ol class="qandadiv" id="fwk-redden-ch08_s02_qs01_qd02_qd01" start="41">
<p class="para" id="fwk-redden-ch08_s02_qs01_p86"><strong class="emphasis bold">Rewrite in standard form and graph.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p87"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0527" display="inline"><mrow><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>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p89"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0529" 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>10</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p91"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0531" 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>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_s02_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p93"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0533" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>14</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>+</mo><mn>40</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p95"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0535" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>6</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_s02_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p97"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0537" 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>20</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p99"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0539" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>12</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_s02_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p101"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0541" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>20</mn><mi>x</mi><mo>−</mo><mn>18</mn><mi>y</mi><mo>+</mo><mn>172</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p103"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0543" 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>8</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_s02_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p105"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0545" display="inline"><mrow><mn>9</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>18</mn><mi>x</mi><mo>+</mo><mn>6</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_s02_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p107"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0547" display="inline"><mrow><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>8</mn><mi>y</mi><mo>+</mo><mn>14</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p109"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0549" 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>4</mn><mi>y</mi><mo>−</mo><mn>15</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p111"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0551" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><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_s02_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p113"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0553" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p115"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0555" 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>8</mn><mi>x</mi><mo>−</mo><mn>12</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_s02_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p117"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0557" display="inline"><mrow><mn>9</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>12</mn><mi>x</mi><mo>−</mo><mn>36</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_s02_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p119"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0559" display="inline"><mrow><mn>2</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>6</mn><mi>x</mi><mo>+</mo><mn>10</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_s02_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p121"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0561" display="inline"><mrow><mn>9</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>6</mn><mi>x</mi><mo>+</mo><mn>12</mn><mi>y</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s02_qs01_qd02_qd02" start="59">
<p class="para" id="fwk-redden-ch08_s02_qs01_p123"><strong class="emphasis bold">Given a circle in general form, determine the intercepts.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p124"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0563" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>6</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p126"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0566" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>2</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_s02_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p128"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0569" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>y</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p130"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0573" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>8</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_s02_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p132"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0577" display="inline"><mrow><mn>2</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>3</mn><mi>x</mi><mo>−</mo><mn>9</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p134"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0581" display="inline"><mrow><mn>3</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>8</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_s02_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p136">Determine the area of the circle whose equation is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0585" 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>6</mn><mi>y</mi><mo>−</mo><mn>35</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p138">Determine the area of the circle whose equation is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0587" 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>12</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>−</mo><mn>59</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p140">Determine the circumference of a circle whose equation is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0589" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p142">Determine the circumference of a circle whose equation is <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0591" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p144">Find general form of the equation of a circle centered at <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0593" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span> passing through <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0594" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p146">Find general form of the equation of a circle centered at <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0596" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span> passing through <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0597" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s02_qs01_qd02_qd03" start="71">
<p class="para" id="fwk-redden-ch08_s02_qs01_p148"><strong class="emphasis bold">Given the graph of a circle, determine its equation in general form.</strong></p>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa71">
<div class="question">
<div class="informalfigure large">
<img src="section_11/d58989bbceae635feae884fc627e2d6f.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa72">
<div class="question">
<div class="informalfigure large">
<img src="section_11/ac4670e7693347253f95f2b55c7d4495.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa73">
<div class="question">
<div class="informalfigure large">
<img src="section_11/56b4affdb27a77635196de9d70d36de9.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa74">
<div class="question">
<div class="informalfigure large">
<img src="section_11/d7d04f2978735b432827ec1e0cf4c784.png">
</div>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch08_s02_qs01_qd03">
<h3 class="title">Part C: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch08_s02_qs01_qd03_qd01" start="75">
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p157">Is the center of a circle part of the graph? Explain.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p158">Make up your own circle, write it in general form, and graph it.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p159">Explain how we can tell the difference between the equation of a parabola in general form and the equation of a circle in general form. Give an example.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch08_s02_qs01_p160">Do all circles have intercepts? What are the possible numbers of intercepts? Illustrate your explanation with graphs.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch08_s02_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p03_ans">Center: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0436" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>,</mo><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span>; radius: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0437" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_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_s02_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p07_ans">Center: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0442" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></math></span>; radius: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0443" display="inline"><mrow><mi>r</mi><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_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_s02_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p11_ans">Center: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0448" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span>; radius: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0449" display="inline"><mrow><mi>r</mi><mo>=</mo><msqrt><mn>7</mn></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_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_s02_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p16_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0455" display="inline"><mrow><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><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>49</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_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_s02_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p20_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0461" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>11</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_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_s02_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p24_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0467" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><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>20</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_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_s02_qs01_qa13_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/53c5e29f570a7738c758a58ac5834001.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_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_s02_qs01_qa15_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/d8ca57b8d216c1ccebcd3aed24774265.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa16_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_s02_qs01_qa17_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/414b9385a66e95450f153f8f70fd8347.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa18_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_s02_qs01_qa19_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/72bbc5fe85bf46b74b342e23182d5b05.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa20_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_s02_qs01_qa21_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/c576480019b7b5be2772d1baf42b8b47.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa22_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_s02_qs01_qa23_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_11/fdbbf0a08c75ade1e4836f8a61e247a1.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa24_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_s02_qs01_qa25_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p54_ans"><em class="emphasis">x</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0484" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>±</mo><msqrt><mn>5</mn></msqrt><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_m0485" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>2</mn><mo>±</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa26_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_s02_qs01_qa27_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p58_ans"><em class="emphasis">x</em>-intercepts: none; <em class="emphasis">y</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0491" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0492" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa28_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_s02_qs01_qa29_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p62_ans"><em class="emphasis">x</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0497" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>±</mo><mn>5</mn><msqrt><mn>2</mn></msqrt><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_m0498" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mo>±</mo><mn>5</mn><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa30_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_s02_qs01_qa31_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p66_ans"><em class="emphasis">x</em>-intercepts: none; <em class="emphasis">y</em>-intercepts: none</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa32_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_s02_qs01_qa33_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p71_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0507" display="inline"><mrow><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><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><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa34_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_s02_qs01_qa35_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p75_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0513" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa36_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_s02_qs01_qa37_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p79_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0519" display="inline"><mrow><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><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><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa38_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_s02_qs01_qa39_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p83_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0524" display="inline"><mrow><mn>7</mn><mi>π</mi></mrow></math></span> square units</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa40_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>
</ol>
<ol class="qandadiv" start="41">
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa41_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p88_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0528" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><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>9</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/5e1fbb14ecec7b5ca105b58313828599.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa42_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_s02_qs01_qa43_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p92_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0532" display="inline"><mrow><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><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/6f5d4f2cbaf1945b4a4423b2ec64d872.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa44_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_s02_qs01_qa45_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p96_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0536" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><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>4</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/05c8f7e1253d32566a6a99ba463beeb7.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa46_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_s02_qs01_qa47_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p100_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0540" display="inline"><mrow><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><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>36</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/3302249100ed24dfc381fedd2e430e62.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa48_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_s02_qs01_qa49_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p104_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0544" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><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>1</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/601a144c1f408d12a7b9d1dbd5bd5049.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa50_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_s02_qs01_qa51_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p108_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0548" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>6</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/f638d834f87348732227d10536c79d5a.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa52_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_s02_qs01_qa53_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p112_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0552" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><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><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/ff4198ff8cd104ae290e7b5c43cec796.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa54_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_s02_qs01_qa55_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p116_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0556" display="inline"><mrow><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><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>2</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/88223a572c933205ba5142e293736b76.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa56_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_s02_qs01_qa57_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p120_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch08_m0560" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>4</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_11/c4b71967d14c60edfc378926c318db75.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa58_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_s02_qs01_qa59_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p125_ans"><em class="emphasis">x</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0564" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0565" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>; <em class="emphasis">y</em>-intercepts: none</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa60_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_s02_qs01_qa61_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch08_s02_qs01_p129_ans"><em class="emphasis">x</em>-intercepts: <span class="inlineequation"><math xml:id="fwk-redden-ch08_m0570" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</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_m0571" 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_m0572" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch08_s02_qs01_qa62_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_s02_qs01_qa63_ans">