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<!DOCTYPE html>
<html>
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<meta charset="UTF-8">
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<title>Using Transformations to Graph Functions</title>
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<div id="book-content">
<div class="section" id="fwk-redden-ch02_s05" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">2.5</span> Using Transformations to Graph Functions</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch02_s05_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch02_s05_o01" numeration="arabic">
<li>Define the rigid transformations and use them to sketch graphs.</li>
<li>Define the non-rigid transformations and use them to sketch graphs.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch02_s05_s01" version="5.0" lang="en">
<h2 class="title editable block">Vertical and Horizontal Translations</h2>
<p class="para editable block" id="fwk-redden-ch02_s05_s01_p01">When the graph of a function is changed in appearance and/or location we call it a transformation. There are two types of transformations. A <span class="margin_term"><a class="glossterm">rigid transformation</a><span class="glossdef">A set of operations that change the location of a graph in a coordinate plane but leave the size and shape unchanged.</span></span> changes the location of the function in a coordinate plane, but leaves the size and shape of the graph unchanged. A <span class="margin_term"><a class="glossterm">non-rigid transformation</a><span class="glossdef">A set of operations that change the size and/or shape of a graph in a coordinate plane.</span></span> changes the size and/or shape of the graph.</p>
<p class="para block" id="fwk-redden-ch02_s05_s01_p02">A <span class="margin_term"><a class="glossterm">vertical translation</a><span class="glossdef">A rigid transformation that shifts a graph up or down.</span></span> is a rigid transformation that shifts a graph up or down relative to the original graph. This occurs when a constant is added to any function. If we add a positive constant to each <em class="emphasis">y</em>-coordinate, the graph will shift up. If we add a negative constant, the graph will shift down. For example, consider the functions <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1329" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1330" display="inline"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow><mo>.</mo></math></span> Begin by evaluating for some values of the independent variable <em class="emphasis">x</em>.</p>
<div class="informalfigure large block">
<img src="section_05/b50310f33dc72e2d8cf74e6611038542.png">
</div>
<p class="para block" id="fwk-redden-ch02_s05_s01_p04">Now plot the points and compare the graphs of the functions <em class="emphasis">g</em> and <em class="emphasis">h</em> to the basic graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1331" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span>, which is shown using a dashed grey curve below.</p>
<div class="informalfigure large block">
<img src="section_05/7e49406ad152f960db8d4ac5ba5f1f28.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s05_s01_p06">The function <em class="emphasis">g</em> shifts the basic graph down 3 units and the function <em class="emphasis">h</em> shifts the basic graph up 3 units. In general, this describes the vertical translations; if <em class="emphasis">k</em> is any positive real number:</p>
<p class="para block" id="fwk-redden-ch02_s05_s01_p07"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Vertical shift up <em class="emphasis">k</em> units:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1332" display="inline"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>k</mi></mrow></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Vertical shift down <em class="emphasis">k</em> units:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1333" display="inline"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>k</mi></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<div class="callout block" id="fwk-redden-ch02_s05_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch02_s05_s01_p08">Sketch the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1334" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s05_s01_p09">Begin with the basic function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1335" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mi>x</mi></msqrt></mrow></math></span> and shift the graph up 4 units.</p>
<p class="para" id="fwk-redden-ch02_s05_s01_p10">Answer: </p>
<div class="informalfigure large">
<img src="section_05/3df380820cd319e9c51a4167305113a3.png">
</div>
</div>
<p class="para block" id="fwk-redden-ch02_s05_s01_p11">A <span class="margin_term"><a class="glossterm">horizontal translation</a><span class="glossdef">A rigid transformation that shifts a graph left or right.</span></span> is a rigid transformation that shifts a graph left or right relative to the original graph. This occurs when we add or subtract constants from the <em class="emphasis">x</em>-coordinate before the function is applied. For example, consider the functions defined by <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1336" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1337" display="inline"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></math></span> and create the following tables:</p>
<div class="informalfigure large block">
<img src="section_05/0bf95c8808240fa0eb0af30d5d71c444.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s05_s01_p13">Here we add and subtract from the <em class="emphasis">x</em>-coordinates and then square the result. This produces a horizontal translation.</p>
<div class="informalfigure large block">
<img src="section_05/533912b244dd5ed2245cdf0b4d869919.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s05_s01_p15">Note that this is the opposite of what you might expect. In general, this describes the horizontal translations; if <em class="emphasis">h</em> is any positive real number:</p>
<p class="para block" id="fwk-redden-ch02_s05_s01_p16"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Horizontal shift left <em class="emphasis">h</em> units:</strong></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1338" display="inline"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Horizontal shift right <em class="emphasis">h</em> units:</strong></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1339" display="inline"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>h</mi><mo stretchy="false">)</mo></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<div class="callout block" id="fwk-redden-ch02_s05_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch02_s05_s01_p17">Sketch the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1340" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s05_s01_p18">Begin with a basic cubing function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1341" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></math></span> and shift the graph 4 units to the right.</p>
<p class="para" id="fwk-redden-ch02_s05_s01_p19">Answer: </p>
<div class="informalfigure large">
<img src="section_05/3b72891ab70b222a56072deb5558b196.png">
</div>
</div>
<p class="para editable block" id="fwk-redden-ch02_s05_s01_p20">It is often the case that combinations of translations occur.</p>
<div class="callout block" id="fwk-redden-ch02_s05_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch02_s05_s01_p21">Sketch the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1342" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>|</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo>|</mo><mo>−</mo><mn>5</mn><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s05_s01_p22">Start with the absolute value function and apply the following transformations.</p>
<p class="para" id="fwk-redden-ch02_s05_s01_p23"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1343" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="left"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>B</mi><mi>a</mi><mi>s</mi><mi>i</mi><mi>c</mi><mtext> </mtext><mi>f</mi><mi>u</mi><mi>n</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo>|</mo></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>H</mi><mi>o</mi><mi>r</mi><mi>i</mi><mi>z</mi><mi>o</mi><mi>n</mi><mi>t</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>s</mi><mi>h</mi><mi>i</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mi>l</mi><mi>e</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mn>3</mn><mtext> </mtext><mi>u</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>s</mi></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo>|</mo><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>V</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>s</mi><mi>h</mi><mi>i</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mi>d</mi><mi>o</mi><mi>w</mi><mi>n</mi><mtext> </mtext><mn>5</mn><mtext> </mtext><mi>u</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>s</mi></mrow></mstyle></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch02_s05_s01_p24">Answer: </p>
<div class="informalfigure large">
<img src="section_05/6e27745f375fee6cc6c806ddd8a66aa4.png">
</div>
</div>
<p class="para editable block" id="fwk-redden-ch02_s05_s01_p25">The order in which we apply horizontal and vertical translations does not affect the final graph.</p>
<div class="callout block" id="fwk-redden-ch02_s05_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch02_s05_s01_p26">Sketch the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1344" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac><mo>+</mo><mn>3</mn><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s05_s01_p27">Begin with the reciprocal function and identify the translations.</p>
<p class="para" id="fwk-redden-ch02_s05_s01_p28"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1345" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="left"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>B</mi><mi>a</mi><mi>s</mi><mi>i</mi><mi>c</mi><mtext> </mtext><mi>f</mi><mi>u</mi><mi>n</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>H</mi><mi>o</mi><mi>r</mi><mi>i</mi><mi>z</mi><mi>o</mi><mi>n</mi><mi>t</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>s</mi><mi>h</mi><mi>i</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mi>r</mi><mi>i</mi><mi>g</mi><mi>h</mi><mi>t</mi><mtext> </mtext><mn>5</mn><mtext> </mtext><mi>u</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>s</mi></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac><mo>+</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>V</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>s</mi><mi>h</mi><mi>i</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mi>u</mi><mi>p</mi><mtext> </mtext><mn>3</mn><mtext> </mtext><mi>u</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>s</mi></mrow></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s05_s01_p29">Take care to shift the vertical asymptote from the <em class="emphasis">y</em>-axis 5 units to the right and shift the horizontal asymptote from the <em class="emphasis">x</em>-axis up 3 units.</p>
<p class="para" id="fwk-redden-ch02_s05_s01_p30">Answer: </p>
<div class="informalfigure large">
<img src="section_05/a1988e97475380515dc5e8a50429f504.png">
</div>
</div>
<div class="callout block" id="fwk-redden-ch02_s05_s01_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch02_s05_s01_p31"><strong class="emphasis bold">Try this!</strong> Sketch the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1346" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s05_s01_p32">Answer:</p>
<div class="informalfigure large">
<img src="section_05/fdb88019f624f31c5c3ca2123fe0b310.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/6F6zKaogxTE" condition="http://img.youtube.com/vi/6F6zKaogxTE/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/6F6zKaogxTE" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch02_s05_s02" version="5.0" lang="en">
<h2 class="title editable block">Reflections</h2>
<p class="para block" id="fwk-redden-ch02_s05_s02_p01">A <span class="margin_term"><a class="glossterm">reflection</a><span class="glossdef">A transformation that produces a mirror image of the graph about an axis.</span></span> is a transformation in which a mirror image of the graph is produced about an axis. In this section, we will consider reflections about the <em class="emphasis">x</em>- and <em class="emphasis">y</em>-axis. The graph of a function is reflected about the <em class="emphasis">x-</em>axis if each <em class="emphasis">y</em>-coordinate is multiplied by −1. The graph of a function is reflected about the <em class="emphasis">y</em>-axis if each <em class="emphasis">x</em>-coordinate is multiplied by −1 before the function is applied. For example, consider <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1347" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mo>−</mo><mi>x</mi></mrow></msqrt></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1348" display="inline"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msqrt><mi>x</mi></msqrt><mo>.</mo></math></span></p>
<div class="informalfigure large block">
<img src="section_05/976a3f92423b4cf72a0331143bb8f61c.png">
</div>
<p class="para block" id="fwk-redden-ch02_s05_s02_p03">Compare the graph of <em class="emphasis">g</em> and <em class="emphasis">h</em> to the basic square root function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1349" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mi>x</mi></msqrt></math></span>, shown dashed in grey below:</p>
<div class="informalfigure large block">
<img src="section_05/dc07456a7de88dacdbfc7d45fb6eb7ad.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s05_s02_p05">The first function <em class="emphasis">g</em> has a negative factor that appears “inside” the function; this produces a reflection about the <em class="emphasis">y</em>-axis. The second function <em class="emphasis">h</em> has a negative factor that appears “outside” the function; this produces a reflection about the <em class="emphasis">x</em>-axis. In general, it is true that:</p>
<p class="para block" id="fwk-redden-ch02_s05_s02_p06"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Reflection about the <em class="emphasis">y</em>-axis:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1350" display="inline"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Reflection about the <em class="emphasis">x</em>-axis:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1351" display="inline"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s05_s02_p07">When sketching graphs that involve a reflection, consider the reflection first and then apply the vertical and/or horizontal translations.</p>
<div class="callout block" id="fwk-redden-ch02_s05_s02_n01">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch02_s05_s02_p08">Sketch the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1352" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s05_s02_p09">Begin with the squaring function and then identify the transformations starting with any reflections.</p>
<p class="para" id="fwk-redden-ch02_s05_s02_p10"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1353" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="left"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>B</mi><mi>a</mi><mi>s</mi><mi>i</mi><mi>c</mi><mtext> </mtext><mi>f</mi><mi>u</mi><mi>n</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>R</mi><mi>e</mi><mi>f</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mi>a</mi><mi>b</mi><mi>o</mi><mi>u</mi><mi>t</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>x</mi><mtext>-</mtext><mi>a</mi><mi>x</mi><mi>i</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>H</mi><mi>o</mi><mi>r</mi><mi>i</mi><mi>z</mi><mi>o</mi><mi>n</mi><mi>t</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>s</mi><mi>h</mi><mi>i</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mi>l</mi><mi>e</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mn>5</mn><mtext> </mtext><mi>u</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>V</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>s</mi><mi>h</mi><mi>i</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mi>u</mi><mi>p</mi><mtext> </mtext><mn>3</mn><mtext> </mtext><mi>u</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch02_s05_s02_p11">Use these translations to sketch the graph.</p>
<p class="para" id="fwk-redden-ch02_s05_s02_p12">Answer:</p>
<div class="informalfigure large">
<img src="section_05/518e388c5d95ac5ef23ad9370e01f647.png">
</div>
</div>
<div class="callout block" id="fwk-redden-ch02_s05_s02_n01a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch02_s05_s02_p13"><strong class="emphasis bold">Try this!</strong> Sketch the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1354" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mo>|</mo><mi>x</mi><mo>|</mo><mo>+</mo><mn>3</mn><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s05_s02_p14">Answer: </p>
<div class="informalfigure large">
<img src="section_05/4bed66ca7372e8e1c8ba5ee4a8047564.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/XsbLkFWWzBc" condition="http://img.youtube.com/vi/XsbLkFWWzBc/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/XsbLkFWWzBc" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch02_s05_s03" version="5.0" lang="en">
<h2 class="title editable block">Dilations</h2>
<p class="para block" id="fwk-redden-ch02_s05_s03_p01">Horizontal and vertical translations, as well as reflections, are called rigid transformations because the shape of the basic graph is left unchanged, or rigid. Functions that are multiplied by a real number other than 1, depending on the real number, appear to be stretched vertically or stretched horizontally. This type of non-rigid transformation is called a <span class="margin_term"><a class="glossterm">dilation</a><span class="glossdef">A non-rigid transformation, produced by multiplying functions by a nonzero real number, which appears to stretch the graph either vertically or horizontally.</span></span>. For example, we can multiply the squaring function <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1355" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></math></span> by 4 and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1356" display="inline"><mfrac><mn>1</mn><mn>4</mn></mfrac></math></span> to see what happens to the graph.</p>
<div class="informalfigure large block">
<img src="section_05/9d3ac7370e44e9f047e4a787962fc4ae.png">
</div>
<p class="para block" id="fwk-redden-ch02_s05_s03_p03">Compare the graph of <em class="emphasis">g</em> and <em class="emphasis">h</em> to the basic squaring function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1357" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></math></span>, shown dashed in grey below:</p>
<div class="informalfigure large block">
<img src="section_05/205680db2491940ca84a05d3de631554.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s05_s03_p05">The function <em class="emphasis">g</em> is steeper than the basic squaring function and its graph appears to have been stretched vertically. The function <em class="emphasis">h</em> is not as steep as the basic squaring function and appears to have been stretched horizontally.</p>
<p class="para editable block" id="fwk-redden-ch02_s05_s03_p06">In general, we have:</p>
<p class="para block" id="fwk-redden-ch02_s05_s03_p07"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Dilation:</strong></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1358" display="inline"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>⋅</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s05_s03_p08">If the factor <em class="emphasis">a</em> is a nonzero fraction between −1 and 1, it will stretch the graph horizontally. Otherwise, the graph will be stretched vertically. If the factor <em class="emphasis">a</em> is negative, then it will produce a reflection as well.</p>
<div class="callout block" id="fwk-redden-ch02_s05_s03_n01">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch02_s05_s03_p09">Sketch the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1359" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>2</mn><mo>|</mo><mi>x</mi><mo>−</mo><mn>5</mn><mo>|</mo><mo>−</mo><mn>3</mn><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s05_s03_p10">Here we begin with the product of −2 and the basic absolute value function: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1360" display="inline"><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mo>|</mo><mi>x</mi><mo>|</mo><mo>.</mo></math></span> This results in a reflection and a dilation.</p>
<p class="para" id="fwk-redden-ch02_s05_s03_p11"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1361" display="block"><mrow><mtable columnlines="solid solid none none" rowlines="solid none none" columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="center"><mi>x</mi></mtd><mtd columnalign="center"><mi>y</mi></mtd><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mo>|</mo><mi>x</mi><mo>|</mo><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>←</mo><mi>D</mi><mi>i</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>r</mi><mi>e</mi><mi>f</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle></mrow></mtd><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mo>|</mo><mo>−</mo><mn>1</mn><mo>|</mo><mo>=</mo><mo>−</mo><mn>2</mn><mo>⋅</mo><mn>1</mn><mo>=</mo><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="right"><mstyle color="#007fbf"><mn>0</mn></mstyle></mtd><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mo>|</mo><mn>0</mn><mo>|</mo><mo>=</mo><mo>−</mo><mn>2</mn><mo>⋅</mo><mn>0</mn><mo>=</mo><mstyle color="#007fbf"><mn>0</mn></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>1</mn></mtd><mtd columnalign="right"><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle></mtd><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mo>|</mo><mn>1</mn><mo>|</mo><mo>=</mo><mo>−</mo><mn>2</mn><mo>⋅</mo><mn>1</mn><mo>=</mo><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch02_s05_s03_p12">Use the points {(−1, −2), (0, 0), (1, −2)} to graph the reflected and dilated function <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1362" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>.</mo></math></span> Then translate this graph 5 units to the right and 3 units down.</p>
<p class="para" id="fwk-redden-ch02_s05_s03_p13"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1363" display="block"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>2</mn><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>B</mi><mi>a</mi><mi>s</mi><mi>i</mi><mi>c</mi><mtext> </mtext><mi>g</mi><mi>r</mi><mi>a</mi><mi>p</mi><mi>h</mi><mtext> </mtext><mi>w</mi><mi>i</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>d</mi><mi>i</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>r</mi><mi>e</mi><mi>f</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mi>a</mi><mi>b</mi><mi>o</mi><mi>u</mi><mi>t</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>x</mi><mo>−</mo><mi>a</mi><mi>x</mi><mi>i</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>2</mn><mo>|</mo><mi>x</mi><mo>−</mo><mn>5</mn><mo>|</mo></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>S</mi><mi>h</mi><mi>i</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mi>r</mi><mi>i</mi><mi>g</mi><mi>h</mi><mi>t</mi><mtext> </mtext><mn>5</mn><mtext> </mtext><mi>u</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>2</mn><mo>|</mo><mi>x</mi><mo>−</mo><mn>5</mn><mo>|</mo><mo>−</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>S</mi><mi>h</mi><mi>i</mi><mi>f</mi><mi>t</mi><mtext> </mtext><mi>d</mi><mi>o</mi><mi>w</mi><mi>n</mi><mtext> </mtext><mn>3</mn><mtext> </mtext><mi>u</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch02_s05_s03_p14">Answer: </p>
<div class="informalfigure large">
<img src="section_05/4df8b5288e81e180e07bb844d15806c8.png">
</div>
</div>
<p class="para editable block" id="fwk-redden-ch02_s05_s03_p15">In summary, given positive real numbers <em class="emphasis">h</em> and <em class="emphasis">k</em>:</p>
<p class="para block" id="fwk-redden-ch02_s05_s03_p16"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Vertical shift up <em class="emphasis">k</em> units:</strong></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1364" display="inline"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>k</mi></mrow></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Vertical shift down <em class="emphasis">k</em> units:</strong></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1365" display="inline"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>k</mi></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block" id="fwk-redden-ch02_s05_s03_p17"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Horizontal shift left <em class="emphasis">h</em> units:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1366" display="inline"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Horizontal shift right <em class="emphasis">h</em> units:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1367" display="inline"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block" id="fwk-redden-ch02_s05_s03_p18"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Reflection about the <em class="emphasis">y</em>-axis:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1368" display="inline"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math></span></p></td>
</tr>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Reflection about the <em class="emphasis">x</em>-axis:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1369" display="inline"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block" id="fwk-redden-ch02_s05_s03_p19"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="right"><p class="para"><strong class="emphasis bold">Dilation:</strong></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1370" display="inline"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>⋅</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<div class="key_takeaways block" id="fwk-redden-ch02_s05_s03_n02">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch02_s05_s03_l01" mark="bullet">
<li>Identifying transformations allows us to quickly sketch the graph of functions. This skill will be useful as we progress in our study of mathematics. Often a geometric understanding of a problem will lead to a more elegant solution.</li>
<li>If a positive constant is added to a function, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1371" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>k</mi></mrow></math></span>, the graph will shift up. If a positive constant is subtracted from a function, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1372" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>k</mi></mrow></math></span>, the graph will shift down. The basic shape of the graph will remain the same.</li>
<li>If a positive constant is added to the value in the domain before the function is applied, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1373" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></math></span>, the graph will shift to the left. If a positive constant is subtracted from the value in the domain before the function is applied, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1374" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></math></span>, the graph will shift right. The basic shape will remain the same.</li>
<li>Multiplying a function by a negative constant, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1375" display="inline"><mrow><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math></span>, reflects its graph in the <em class="emphasis">x</em>-axis. Multiplying the values in the domain by −1 before applying the function, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1376" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math></span>, reflects the graph about the <em class="emphasis">y</em>-axis.</li>
<li>When applying multiple transformations, apply reflections first.</li>
<li>Multiplying a function by a constant other than 1, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1377" display="inline"><mrow><mi>a</mi><mo>⋅</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></math></span>, produces a dilation. If the constant is a positive number greater than 1, the graph will appear to stretch vertically. If the positive constant is a fraction less than 1, the graph will appear to stretch horizontally.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch02_s05_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s05_qs01_qd01">
<h3 class="title">Part A: Vertical and Horizontal Translations</h3>
<p class="para"><strong class="emphasis bold">Match the graph to the function definition.</strong></p>
<div class="informalfigure large">
<img src="section_05/caf84d0d27db512ef90d11b59b6c37dc.png">
</div>
<div class="informalfigure large">
<img src="section_05/2fe54b1c80ea84f0f721462f90455c0b.png">
</div>
<div class="informalfigure large">
<img src="section_05/3622a0d2256166544a122ecd7156de36.png">
</div>
<div class="informalfigure large">
<img src="section_05/47088a9efd6814511cb0fc8d233b539f.png">
</div>
<div class="informalfigure large">
<img src="section_05/d44d62205d34ed371aad179b77c54a81.png">
</div>
<div class="informalfigure large">
<img src="section_05/252deea445e76b98c6412384dd94246d.png">
</div>
<ol class="qandadiv" id="fwk-redden-ch02_s05_qs01_qd01_qd01">
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1378" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1379" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>|</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo>|</mo><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1380" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1381" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>|</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo>|</mo><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1382" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msqrt><mo>+</mo><mn>1</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1383" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>|</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>|</mo><mo>−</mo><mn>2</mn></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s05_qs01_qd01_qd02" start="7">
<p class="para" id="fwk-redden-ch02_s05_qs01_p14"><strong class="emphasis bold">Graph the given function. Identify the basic function and translations used to sketch the graph. Then state the domain and range.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p15"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1384" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1388" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>−</mo><mn>2</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p21"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1392" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p24"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1396" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p27"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1400" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p30"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1404" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p33"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1408" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p36"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1412" display="inline"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p39"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1416" display="inline"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>|</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo>|</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p42"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1420" display="inline"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>|</mo><mi>x</mi><mo>−</mo><mn>4</mn><mo>|</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p45"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1424" display="inline"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>|</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo>|</mo><mo>−</mo><mn>3</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p48"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1428" display="inline"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>|</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>|</mo><mo>−</mo><mn>5</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p51"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1432" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>5</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p54"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1436" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></msqrt></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p57"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1440" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></msqrt><mo>+</mo><mn>1</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p60"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1444" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msqrt><mo>+</mo><mn>3</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p63"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1448" display="inline"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p66"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1452" display="inline"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>4</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p69"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1456" display="inline"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p72"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1460" display="inline"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1464" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p78"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1468" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p81"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1472" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mn>5</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p84"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1476" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>−</mo><mn>3</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p87"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1480" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mn>2</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p90"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1484" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mn>3</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p93"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1488" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>4</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p96"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1491" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p99"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1494" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mroot><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>6</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p102"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1498" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mroot><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>−</mo><mn>4</mn></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s05_qs01_qd01_qd03" start="37">
<p class="para" id="fwk-redden-ch02_s05_qs01_p105"><strong class="emphasis bold">Graph the piecewise functions.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa37">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1502" display="block"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>x</mi><mo><</mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>x</mi><mo>+</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>x</mi><mo>≥</mo><mn>0</mn></mtd></mtr></mtable></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa38">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1503" display="block"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>x</mi><mo><</mo><mn>0</mn></mtd></mtr><mtr><mtd><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>3</mn><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>x</mi><mo>≥</mo><mn>0</mn></mtd></mtr></mtable></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa39">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1504" display="block"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>x</mi><mo><</mo><mn>0</mn></mtd></mtr><mtr><mtd><mo>|</mo><mi>x</mi><mo>−</mo><mn>3</mn><mo>|</mo><mo>−</mo><mn>4</mn><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>x</mi><mo>≥</mo><mn>0</mn></mtd></mtr></mtable></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa40">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1505" display="block"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo stretchy="true">{</mo> <mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mtd><mtd columnalign="left"><mrow><mtext>if</mtext></mrow></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo><</mo><mn>0</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mrow><mtext>if</mtext></mrow></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math>
</span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa41">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1506" display="block"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>x</mi><mo><</mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>2</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>x</mi><mo>≥</mo><mn>0</mn></mtd></mtr></mtable></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa42">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1507" display="block"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mrow><mtext>if</mtext></mrow></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo><</mo><mn>0</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow><mtext>if</mtext></mrow></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa43">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1508" display="block"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>10</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>x</mi><mo><</mo><mo>−</mo><mn>8</mn></mtd></mtr><mtr><mtd><mi>x</mi><mo>+</mo><mn>4</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mo>−</mo><mn>8</mn><mo>≤</mo><mi>x</mi><mo><</mo><mo>−</mo><mn>4</mn></mtd></mtr><mtr><mtd><msqrt><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msqrt><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>x</mi><mo>≥</mo><mo>−</mo><mn>4</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa44">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1509" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mi>x</mi><mo>+</mo><mn>10</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>x</mi><mo>≤</mo><mo>−</mo><mn>10</mn></mtd></mtr><mtr><mtd><mo>|</mo><mi>x</mi><mo>−</mo><mn>5</mn><mo>|</mo><mo>−</mo><mn>15</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mo>−</mo><mn>10</mn><mo><</mo><mi>x</mi><mo>≤</mo><mn>20</mn></mtd></mtr><mtr><mtd><mn>10</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>if</mtext><mtext> </mtext><mi>x</mi><mo>></mo><mn>20</mn></mtd></mtr></mtable></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s05_qs01_qd01_qd04" start="45">
<p class="para" id="fwk-redden-ch02_s05_qs01_p122"><strong class="emphasis bold">Write an equation that represents the function whose graph is given.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa45">
<div class="question">
<div class="informalfigure large">
<img src="section_05/9de0ccbee5d59fa05acab85d085cb4d7.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa46">
<div class="question">
<div class="informalfigure large">
<img src="section_05/122abd4ccb8eb72a59532b22ed6116ab.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa47">
<div class="question">
<div class="informalfigure large">
<img src="section_05/b8f5c01476fe7ee9f21dd781da420d2d.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa48">
<div class="question">
<div class="informalfigure large">
<img src="section_05/55503ba3d165669977cae857ebddb4e0.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa49">
<div class="question">
<div class="informalfigure large">
<img src="section_05/e90ad3255312e9bf25d0f866de703eb4.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa50">
<div class="question">
<div class="informalfigure large">
<img src="section_05/6c160b69a9ef56763a5424ea14fbc86f.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa51">
<div class="question">
<div class="informalfigure large">
<img src="section_05/613f12af91bfbf853201387cb6dd7acb.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa52">
<div class="question">
<div class="informalfigure large">
<img src="section_05/dc68a19f31725aae971d758ae3db642f.png">
</div>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s05_qs01_qd02">
<h3 class="title">Part B: Reflections and Dilations</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s05_qs01_qd02_qd01" start="53">
<p class="para" id="fwk-redden-ch02_s05_qs01_p139"><strong class="emphasis bold">Match the graph the given function definition.</strong></p>
<div class="informalfigure large">
<img src="section_05/19f3c208cfdeeffde7e76281b4b28f46.png">
</div>
<div class="informalfigure large">
<img src="section_05/7ddabfc77a72214e9f6bea00e3b2cca0.png">
</div>
<div class="informalfigure large">
<img src="section_05/039e6f4a86d07a578660882bccf7ea40.png">
</div>
<div class="informalfigure large">
<img src="section_05/16b19343fd01aecf51c1cdea8af3ee21.png">
</div>
<div class="informalfigure large">
<img src="section_05/26cdff42b4eb188a4512c934fd59f9e5.png">
</div>
<div class="informalfigure large">
<img src="section_05/75295519ff6aaa13dced0dc6ed6e2ef7.png">
</div>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p141"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1518" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>3</mn><mo>|</mo><mi>x</mi><mo>|</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p143"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1519" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p145"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1520" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mo>|</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>|</mo><mo>+</mo><mn>2</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p147"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1521" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p149"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1522" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>|</mo><mi>x</mi><mo>|</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p151"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1523" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s05_qs01_qd02_qd02" start="59">
<p class="para" id="fwk-redden-ch02_s05_qs01_p153"><strong class="emphasis bold">Use the transformations to graph the following functions.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p154"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1524" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>x</mi><mo>+</mo><mn>5</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p156"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1525" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mo>|</mo><mi>x</mi><mo>|</mo><mo>−</mo><mn>3</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p158"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1526" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mo>|</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo>|</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p160"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1527" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p162"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1528" display="inline"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mo>−</mo><mi>x</mi></mrow></msqrt><mo>+</mo><mn>2</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p164"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1529" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>2</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p166"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1530" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><msup><mo stretchy="false">)</mo><mn>3</mn></msup></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p168"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1531" display="inline"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msqrt><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></msqrt><mo>+</mo><mn>1</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p170"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1532" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>4</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p172"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1533" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p174"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1534" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>3</mn><mo>|</mo><mi>x</mi><mo>|</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p176"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1535" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p178"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1536" display="inline"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p180"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1537" display="inline"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p182"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1538" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msqrt><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p184"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1539" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>5</mn><msqrt><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msqrt></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p186"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1540" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><msqrt><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>+</mo><mn>2</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p188"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1541" display="inline"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p190"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1542" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>3</mn><msup><mo stretchy="false">)</mo><mn>3</mn></msup><mo>−</mo><mn>1</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p192"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1543" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>5</mn><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>3</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mn>3</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p194"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1544" display="inline"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>3</mn><mo>|</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo>|</mo><mo>−</mo><mn>2</mn></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p196"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1545" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p198"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1546" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p200"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1547" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mn>2</mn></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s05_qs01_qd03">
<h3 class="title">Part C: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s05_qs01_qd03_qd01" start="83">
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p202">Use different colors to graph the family of graphs defined by <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1548" display="inline"><mi>y</mi><mo>=</mo><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1549" display="inline"><mi>k</mi><mo>∈</mo><mrow><mo>{</mo><mrow><mn>1</mn><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo>}</mo></mrow><mo>.</mo></math></span> What happens to the graph when the denominator of <em class="emphasis">k</em> is very large? Share your findings on the discussion board.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p203">Graph <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1550" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mi>x</mi></msqrt></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1551" display="inline"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msqrt><mi>x</mi></msqrt></math></span> on the same set of coordinate axes. What does the general shape look like? Try to find a single equation that describes the shape. Share your findings.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch02_s05_qs01_p204">Explore what happens to the graph of a function when the domain values are multiplied by a factor <em class="emphasis">a</em> before the function is applied, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1552" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo></math></span> Develop some rules for this situation and share them on the discussion board.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch02_s05_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch02_s05_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch02_s05_qs01_p03_ans">e</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch02_s05_qs01_p07_ans">d</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch02_s05_qs01_p11_ans">f</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa07_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/ed14f13811bfb7c397b768ab1e6d718a.png">
</div>
<p class="para" id="fwk-redden-ch02_s05_qs01_p17_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1385" display="inline"><mi>y</mi><mo>=</mo><mi>x</mi></math></span>; Shift up 3 units; domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1386" display="inline"><mi>ℝ</mi></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1387" display="inline"><mi>ℝ</mi></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa09_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/0e393f0d6e151259a123b1e505dec86b.png">
</div>
<p class="para" id="fwk-redden-ch02_s05_qs01_p23_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1393" display="inline"><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span>; Shift up 1 unit; domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1394" display="inline"><mi>ℝ</mi></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1395" display="inline"><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa11_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/8fc7f879a8ba5f12d0b98f348e5adadb.png">
</div>
<p class="para" id="fwk-redden-ch02_s05_qs01_p29_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1401" display="inline"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></math></span>; Shift right 5 units; domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1402" display="inline"><mi>ℝ</mi></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1403" display="inline"><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa13_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/57a5fd7bcf0e225b10961c6534cd4545.png">
</div>
<p class="para" id="fwk-redden-ch02_s05_qs01_p35_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1409" display="inline"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></math></span>; Shift right 5 units and up 2 units; domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1410" display="inline"><mi>ℝ</mi></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1411" display="inline"><mo stretchy="false">[</mo><mn>2</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa15_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/86a1d10b4aad0ab79bc2c8dd55bf4f38.png">
</div>
<p class="para" id="fwk-redden-ch02_s05_qs01_p41_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1417" display="inline"><mi>y</mi><mo>=</mo><mo>|</mo><mi>x</mi><mo>|</mo></math></span>; Shift left 4 units; domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1418" display="inline"><mi>ℝ</mi></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1419" display="inline"><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa17_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/424b66df0df22a96fd88c4957413d44e.png">
</div>
<p class="para" id="fwk-redden-ch02_s05_qs01_p47_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1425" display="inline"><mi>y</mi><mo>=</mo><mo>|</mo><mi>x</mi><mo>|</mo></math></span>; Shift right 1 unit and down 3 units; domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1426" display="inline"><mi>ℝ</mi></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1427" display="inline"><mo stretchy="false">[</mo><mo>−</mo><mn>3</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa19_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/457665c1ea5709240bd4c6e1685a1985.png">
</div>
<p class="para" id="fwk-redden-ch02_s05_qs01_p53_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1433" display="inline"><mi>y</mi><mo>=</mo><msqrt><mi>x</mi></msqrt></math></span>; Shift down 5 units; domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1434" display="inline"><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1435" display="inline"><mo stretchy="false">[</mo><mo>−</mo><mn>5</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa21_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/da6d3f21b303aeb0b29fe4975b48a64f.png">
</div>
<p class="para" id="fwk-redden-ch02_s05_qs01_p59_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1441" display="inline"><mi>y</mi><mo>=</mo><msqrt><mi>x</mi></msqrt></math></span>; Shift right 2 units and up 1 unit; domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1442" display="inline"><mo stretchy="false">[</mo><mn>2</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1443" display="inline"><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa23_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/01b74b05906d95ff14c5aa6de0ae7b4f.png">
</div>
<p class="para" id="fwk-redden-ch02_s05_qs01_p65_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1449" display="inline"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></math></span>; Shift right 2 units; domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1450" display="inline"><mi>ℝ</mi></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1451" display="inline"><mi>ℝ</mi></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa25_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/a4f584febcd95dc5ef92bbe2ef80df7c.png">
</div>
<p class="para" id="fwk-redden-ch02_s05_qs01_p71_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1457" display="inline"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></math></span>; Shift right 1 unit and down 4 units; domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1458" display="inline"><mi>ℝ</mi></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1459" display="inline"><mi>ℝ</mi></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa27_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/75fa23d883d738eeb47a020057002b8f.png">
</div>
<p class="para" id="fwk-redden-ch02_s05_qs01_p77_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1465" display="inline"><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></math></span>; Shift right 2 units; domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1466" display="inline"><mo stretchy="false">(</mo><mo>−</mo><mi>∞</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1467" display="inline"><mo stretchy="false">(</mo><mo>−</mo><mi>∞</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa29_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/53d3a12d61be06d8913ae13668760ebb.png">
</div>
<p class="para" id="fwk-redden-ch02_s05_qs01_p83_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1473" display="inline"><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></math></span>; Shift up 5 units; domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1474" display="inline"><mo stretchy="false">(</mo><mo>−</mo><mi>∞</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1475" display="inline"><mo stretchy="false">(</mo><mo>−</mo><mi>∞</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa31_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/0eac4ad67881e57bfa8e7dc46c933e8e.png">
</div>
<p class="para" id="fwk-redden-ch02_s05_qs01_p89_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1481" display="inline"><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></math></span>; Shift left 1 unit and down 2 units; domain: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1482" display="inline"><mo stretchy="false">(</mo><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span>; range: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1483" display="inline"><mo stretchy="false">(</mo><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>,</mo><mi>∞</mi><mo stretchy="false">)</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s05_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-ch02_s05_qs01_qa33_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/dec428893d68980da985eabaf7f7fb11.png">