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class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../539/functions.html">Functions</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Transforming Functions HL</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/functions/transforming-functions/main-1.jpg" style="float: left; width: 100px; height: 100px;">For the topic of transforming functions, we need to understand the effect of <strong>translating</strong>, <strong>reflecting</strong> and <strong>stretching </strong>has on functions. On this page, we will also consider graphs of <span class="math-tex">\(|f(x)|\)</span> , <span class="math-tex">\(f(|x|)\)</span> , <span class="math-tex">\(\frac{1}{f(x)}\)</span> as well as [f(x)]² . You may be asked to describe the transformations, sketch graphs or find the coordinates of points that have been transformed. Whilst technology can be a big help understanding these transformations, questions often require you to answer them without your graphical calculator.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <p>On this page, you should learn about</p> <ul> <li>transformations of graphs <ul> <li>vertical translations: <span class="math-tex">\(y=f(x)+b\)</span> </li> <li>horizontal translations: <span class="math-tex">\(y=f(x-a)\)</span></li> <li>reflection in x axis: <span class="math-tex">\(y=-f(x)\)</span></li> <li>reflection in y axis: <span class="math-tex">\(y=f(-x)\)</span></li> <li>vertical stretch: <span class="math-tex">\(y=af(x)\)</span></li> <li>horizontal stretch: <span class="math-tex">\(y=f(ax)\)</span></li> </ul> </li> <li>compositions of any of the above transformations</li> <li>the graphs of the functions <ul> <li><span class="math-tex">\(y=|f(x)|\)</span></li> <li><span class="math-tex">\(y=f(|x|)\)</span></li> <li><span class="math-tex">\(y=\frac{1}{f(x)}\)</span></li> <li><span class="math-tex">\(y=[f(x)]^2\)</span></li> <li><span class="math-tex">\(y=f(ax+b)\)</span></li> </ul> </li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <p>These graphs should help you understand the transformations of functions</p> <div class="panel panel-has-colored-body panel-yellow panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Vertical Stretch y = af(x)</p> </div> </div> <div class="panel-body"> <div> <p>Drag the slider to see the effect</p> <p style="text-align: center;"><iframe height="564px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/jnh3vgup/width/538/height/564/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="Vertical Stretch" width="538px"></iframe></p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Horizontal Stretch y = f(ax)</p> </div> </div> <div class="panel-body"> <div> <p>Click on the image to open an applet and explore this transformation:</p> <p><a href="../1803/horizontal-stretch.html" target="_blank"><img alt="" src="../../files/functions/transforming-functions/horizontal-stretch-thumbnail.png" style="width: 300px; height: 327px;"></a></p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Translations y = f(x - a) + b</p> </div> </div> <div class="panel-body"> <div>Click on the image to open an applet and explore this transformation:</div> <div><a href="../1804/translations.html" target="_blank"><img alt="" src="../../files/functions/transforming-functions/translation-thumbnail.png" style="width: 300px; height: 340px;"></a></div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>|f(x)| and f(|x|)</p> </div> </div> <div class="panel-body"> <div> <p>Click on the image to open an applet and explore this transformation:</p> <p><a href="../1805/absolute-functions.html" target="_blank"><img alt="" src="../../files/functions/transforming-functions/absolute-function.png" style="width: 300px; height: 328px;"></a></p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1/f(x)</p> </div> </div> <div class="panel-body"> <div> <p>Click on the image to open an applet and explore this transformation: </p> <p><a href="../1806/1-over-function.html" target="_blank"><img alt="" src="../../files/functions/transforming-functions/1-over-f.png" style="width: 300px; height: 217px;"></a></p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/functions/transforming-functions/revision-notes_transforming_functionshl.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/functions/transforming-functions/revision-notes_transforming_functionshl.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <div> <p>Here is a quiz that practises the transformations af(x) , f(ax), f(x)+a , f(x - a)</p> </div> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#031f2c81"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="031f2c81"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Transforming Functions <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-217-875" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) is transformed using translations and/or stretches. Describe the transformation from y = f(x) in each case below by dragging the description to the correct place</p><p><span class="q-text-draggable draggable" draggable="true">translate 2 units up</span> <span class="q-text-draggable draggable" draggable="true">translate 2 units down</span> <span class="q-text-draggable draggable" draggable="true">translate 2 units right</span> <span class="q-text-draggable draggable" draggable="true">translate 2 units left</span> <span class="q-text-draggable draggable" draggable="true">vertical stretch with scale factor 2</span> <span class="q-text-draggable draggable" draggable="true">horizontal stretch with scale factor 2</span> <span class="q-text-draggable draggable" draggable="true">horizontal stretch with scale factor 0.5</span> <span class="q-text-draggable draggable" draggable="true">vertical stretch with scale factor 0.5</span></p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>f(x) + 2 <input type="text" style="height: auto;" data-c="translate 2 units up"> <span class="review"></span></td><td>2f(x) <input type="text" style="height: auto;" data-c="vertical stretch with scale factor 2"> <span class="review"></span></td></tr><tr><td>f(x - 2) <input type="text" style="height: auto;" data-c="translate 2 units right"> <span class="review"></span></td><td>f(2x) <input type="text" style="height: auto;" data-c="horizontal stretch with scale factor 0.5"> <span class="review"></span></td></tr><tr><td>f(0.5x) <input type="text" style="height: auto;" data-c="horizontal stretch with scale factor 2"> <span class="review"></span></td><td>f(x + 2) <input type="text" style="height: auto;" data-c="translate 2 units left"> <span class="review"></span></td></tr></tbody></table></div><div class="q-explanation"><p>f(x - <strong>a</strong>) is translation<em><strong> a</strong></em> units right</p><p>f(x) +<em><strong> a</strong></em> is a traslation <strong><em>a </em></strong>units up</p><p><em><strong>a</strong></em>f(x) is a vertical stretch with scale factor <strong><em>a</em></strong></p><p>f(<strong><em>a</em></strong>x) is a horizontal stretch with scale factor <em><strong><span class="math-tex">\(\frac{1}{a}\)</span></strong></em></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) is transformed using translations and/or stretches. Describe the transformation from y = f(x) in each case below by dragging the function to the correct place</p><p> <span class="q-text-draggable draggable" draggable="true">f(-x)</span> <span class="q-text-draggable draggable" draggable="true">-f(x)</span> <span class="q-text-draggable draggable" draggable="true">f(4x)</span> <span class="q-text-draggable draggable" draggable="true">4f(x)</span> <span class="q-text-draggable draggable" draggable="true">f(0.25x)</span> <span class="q-text-draggable draggable" draggable="true">f(4x)</span> <span class="q-text-draggable draggable" draggable="true">0.25f(x)</span> <span class="q-text-draggable draggable" draggable="true">f(x + 4)</span> <span class="q-text-draggable draggable" draggable="true">f(x - 4)</span> <span class="q-text-draggable draggable" draggable="true">f(x) + 4</span> <span class="q-text-draggable draggable" draggable="true">f(x) - 4</span></p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Reflection in the y axis <input type="text" style="height: auto;" data-c="f(-x)"> <span class="review"></span></td><td>vertical stretch with scale factor 4 <input type="text" style="height: auto;" data-c="4f(x)"> <span class="review"></span></td></tr><tr><td>horizontal stretch with scale factor 4<input type="text" style="height: auto;" data-c="f(0.25x)"> <span class="review"></span></td><td>translation of <span class="math-tex">\(\left( \begin{matrix} 4 \\ 0 \end{matrix} \right) \)</span> <input type="text" style="height: auto;" data-c="f(x - 4)"> <span class="review"></span></td></tr><tr><td>translation of <span class="math-tex">\(\left( \begin{matrix} 0 \\ -4 \end{matrix} \right) \)</span> <input type="text" style="height: auto;" data-c="f(x) - 4"> <span class="review"></span></td><td>Reflection in the x axis <input type="text" style="height: auto;" data-c="-f(x)"> <span class="review"></span></td></tr></tbody></table></div><div class="q-explanation"><p>f(x - <strong>a</strong>) is translation<em><strong> a</strong></em> units right</p><p>f(x) +<em><strong> a</strong></em> is a traslation <strong><em>a </em></strong>units up</p><p><em><strong>a</strong></em>f(x) is a vertical stretch with scale factor <strong><em>a</em></strong></p><p>f(<strong><em>a</em></strong>x) is a horizontal stretch with scale factor <em><strong><span class="math-tex">\(\frac{1}{a}\)</span></strong></em></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) is transformed. Give the equation of the new graph after these transformations.</p><p>Careful, no stretch is NOT a stretch factor of zero!</p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Vertical stretch scale factor 3<hr class="hidden-separator"></td><td>y = <strong><em>a</em></strong>f(<strong><em>b</em></strong>x)<hr class="hidden-separator"></td><td><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> <strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span><hr class="hidden-separator"></td></tr><tr><td>Translation of <span class="math-tex">\(\left( \begin{matrix} 1 \\ 2 \end{matrix} \right) \)</span><hr class="hidden-separator"></td><td>y = f(x - <strong><em>a</em></strong>) + <strong><em>b</em></strong><hr class="hidden-separator"></td><td><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> <strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span><hr class="hidden-separator"></td></tr><tr><td><p>Vertical stretch factor 2 and</p><p>horizontal stretch scale factor 2</p><hr class="hidden-separator"></td><td>y = <strong><em>a</em></strong>f(<strong><em>b</em></strong>x)<hr class="hidden-separator"></td><td><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> <em><strong>b</strong></em> = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span><hr class="hidden-separator"></td></tr><tr><td><p>Reflection in the y axis and</p><p>vertical stretch scale factor 4</p><hr class="hidden-separator"></td><td>y = <strong><em>a</em></strong>f(<strong><em>b</em></strong>x)<hr class="hidden-separator"></td><td><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> <strong><em>b</em></strong>= <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span><hr class="hidden-separator"></td></tr><tr><td><p>Vertical stretch scale factor 0.5,</p><p>Reflection in the x axis and</p><p>Translation of <span class="math-tex">\(\left( \begin{matrix} -1 \\ -3 \end{matrix} \right) \)</span></p><hr class="hidden-separator"></td><td>y = <strong><em>a</em></strong>f(<strong><em>b</em></strong>x - <strong><em>c</em></strong>) + <strong><em>d</em></strong><hr class="hidden-separator"></td><td><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span> <strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p><strong><em>c</em></strong> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span> <strong><em>d</em></strong> = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p><hr class="hidden-separator"></td></tr><tr><td><p>Reflection in the x axis,</p><p>Horizontal stretch scale factor 0.2 and</p><p>Translation of <span class="math-tex">\(\left( \begin{matrix} 0 \\ 2 \end{matrix} \right) \)</span></p></td><td>y = <strong><em>a</em></strong>f(<strong><em>b</em></strong>x - <strong><em>c</em></strong>) + <strong><em>d</em></strong></td><td><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span> <strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><p><strong><em>c</em></strong> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span> <strong><em>d</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></td></tr></tbody></table></div><div class="q-explanation"><p>f(x - <strong>a</strong>) is translation<em><strong> a</strong></em> units right</p><p>f(x) +<em><strong> a</strong></em> is a traslation <strong><em>a </em></strong>units up</p><p><em><strong>a</strong></em>f(x) is a vertical stretch with scale factor <strong><em>a</em></strong></p><p>f(<strong><em>a</em></strong>x) is a horizontal stretch with scale factor <em><strong><span class="math-tex">\(\frac{1}{a}\)</span></strong></em></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of the function y = f(x) is plotted in blue in each of the graphs below. Match up the functions with the other graph in red</p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">f(x + 2)</span> <span class="q-text-draggable draggable" draggable="true">f(x - 2)</span> <span class="q-text-draggable draggable" draggable="true">-f(x - 2)</span> <span class="q-text-draggable draggable" draggable="true">-f(x) + 2</span> <span class="q-text-draggable draggable" draggable="true">f(x) - 2</span> <span class="q-text-draggable draggable" draggable="true">-f(x) - 2</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><p> y=<input type="text" style="height: auto;" data-c="f(x + 2)"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q4a.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td><td><p>y=<input type="text" style="height: auto;" data-c="f(x - 2)"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q4b.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td><td><p>y=<input type="text" style="height: auto;" data-c="-f(x - 2)"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q4c.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td></tr><tr><td><p>y=<input type="text" style="height: auto;" data-c="f(x) - 2"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q4e.jpg" style="width: 200px; height: 200px;"></p></td><td><p>y=<input type="text" style="height: auto;" data-c="-f(x) + 2"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q4d.jpg" style="width: 200px; height: 200px;"></p></td><td><p>y=<input type="text" style="height: auto;" data-c="-f(x) - 2"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q4f.jpg" style="width: 200px; height: 200px;"></p></td></tr><tr><td></td><td></td><td> </td></tr></tbody></table></div><div class="q-explanation"><p>f(-x) is a reflection in the y axis</p><p>-f(x) is a reflection in the x axis</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of the function y = f(x) is plotted in blue in each of the graphs below. Match up the functions with the other graph in red</p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">-f(x)</span> <span class="q-text-draggable draggable" draggable="true">f(x) - 2</span> <span class="q-text-draggable draggable" draggable="true">f(x - 2)</span> <span class="q-text-draggable draggable" draggable="true">f(-x)</span> <span class="q-text-draggable draggable" draggable="true">-f(-x)</span> <span class="q-text-draggable draggable" draggable="true">-f(x - 2)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><p>y = <input type="text" style="height: auto;" data-c="-f(x)"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q5a.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td><td><p>y = <input type="text" style="height: auto;" data-c="f(x) - 2"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q5b.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td><td><p>y = <input type="text" style="height: auto;" data-c="f(x - 2)"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q5c.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td></tr><tr><td><p>y = <input type="text" style="height: auto;" data-c="f(-x)"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q5d.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td><td><p>y = <input type="text" style="height: auto;" data-c="-f(-x)"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q5e.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td><td><p>y = <input type="text" style="height: auto;" data-c="-f(x - 2)"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q5f.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td></tr></tbody></table></div><div class="q-explanation"><p>f(-x) is a reflection in the y axis</p><p>-f(x) is a reflection in the x axis</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of the function y = f(x) is plotted in blue in each of the graphs below. Match up the functions with the other graph in red</p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">f(x) - 1</span> <span class="q-text-draggable draggable" draggable="true">f(x - 2)</span> <span class="q-text-draggable draggable" draggable="true">f(-x)</span> <span class="q-text-draggable draggable" draggable="true">-f(x)</span> <span class="q-text-draggable draggable" draggable="true">f(x + 2)</span> <span class="q-text-draggable draggable" draggable="true">f(x) + 1</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><p>y = <input type="text" style="height: auto;" data-c="f(x) - 1"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q6a.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td><td><p>y = <input type="text" style="height: auto;" data-c="f(x - 2)"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q6b.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td><td><p>y = <input type="text" style="height: auto;" data-c="f(-x)"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q6c.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td></tr><tr><td><p>y = <input type="text" style="height: auto;" data-c="-f(x)"> <span class="review"></span></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q6d.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td><td><p>y = <input type="text" style="height: auto;" data-c="f(x + 2)"> <span class="review"></span></p><img alt="" src="../../files/functions/transforming-functions/quiz/q6e.jpg" style="width: 200px; height: 200px;"><hr class="hidden-separator"></td><td><p>y = <input type="text" style="height: auto;" data-c="f(x) + 1"> <span class="review"></span></p><img alt="" src="../../files/functions/transforming-functions/quiz/q6f.jpg" style="width: 200px; height: 200px;"><hr class="hidden-separator"></td></tr></tbody></table></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) with the coordinates of A, B and C is shown below</p><p>Give the coordinates of A', B' and C' after the transformation <em><strong>2f(-x)</strong></em></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q7.jpg" style="width: 300px; height: 300px;"></p></div><div class="q-answer"><p>A'(x<sub>A</sub> , y<sub>A</sub>) </p><p>x<sub>A</sub> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> y<sub>A</sub> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p><hr class="hidden-separator"><p>B'(x<sub>B</sub> , y<sub>B</sub>) </p><p>x<sub>B</sub> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span> y<sub>B</sub> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><hr class="hidden-separator"><p>C'(x<sub>C</sub> , y<sub>C</sub>) </p><p>x<sub>C</sub> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> y<sub>C</sub> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p>The transformation is</p><ul><li>Reflection in the y axis </li><li>Vertical stretch scale factor 2</li></ul></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) with the coordinates of A, B and C is shown below</p><p>Give the coordinates of A', B' and C' after the transformation <em><strong>f(-2x) + 1</strong></em></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q7.jpg" style="width: 300px; height: 300px;"></p></div><div class="q-answer"><p>A'(x<sub>A</sub> , y<sub>A</sub>) </p><p>x<sub>A</sub> = <input type="text" style="height: auto;" data-c="1.5"> <span class="review"></span> y<sub>A</sub> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><hr class="hidden-separator"><p>B'(x<sub>B</sub> , y<sub>B</sub>) </p><p>x<sub>B</sub> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span> y<sub>B</sub> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><hr class="hidden-separator"><p>C'(x<sub>C</sub> , y<sub>C</sub>) </p><p>x<sub>C</sub> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span> y<sub>C</sub> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>The transformation is</p><ul><li>Reflection in the x axis </li><li>Horizontal stretch scale factor 0.5</li><li>Translation of <span class="math-tex">\(\left( \begin{matrix} 0 \\ 1 \end{matrix} \right) \)</span></li></ul></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) with the coordinates of A, B and C is shown below</p><p>Give the coordinates of A', B' and C' after the transformation <em><strong>-f(x + 1) - 2</strong></em></p><p><img alt="" src="../../files/functions/transforming-functions/quiz/q7.jpg" style="width: 300px; height: 300px;"></p></div><div class="q-answer"><p>A'(x<sub>A</sub> , y<sub>A</sub>) </p><p>x<sub>A</sub> = <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span> y<sub>A</sub> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p><hr class="hidden-separator"><p>B'(x<sub>B</sub> , y<sub>B</sub>) </p><p>x<sub>B</sub> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span> y<sub>B</sub> = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p><hr class="hidden-separator"><p>C'(x<sub>C</sub> , y<sub>C</sub>) </p><p>x<sub>C</sub> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> y<sub>C</sub> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p>The transformation is</p><ul><li>Translation of <span class="math-tex">\(\left( \begin{matrix} -1 \\ 0 \end{matrix} \right) \)</span></li><li>Reflection in x axis</li><li>Translation of <span class="math-tex">\(\left( \begin{matrix} 0 \\ -2 \end{matrix} \right) \)</span></li></ul></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><span></span></p>Let <em><strong>f </strong></em>and <em><strong>g</strong></em> be the functions such that <em><strong>g</strong></em>(x) = -3<em><strong>f</strong></em>(2x +1) +1<p>The point A (2,5) on the graph of <strong><em>f</em></strong> is mapped to the point A' on the graph of <strong><em>g</em></strong></p><p>Find A'</p></div><div class="q-answer"><p>A'(x<sub>A</sub> , y<sub>A</sub>) </p><p>x<sub>A</sub> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span> y<sub>A</sub> = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span></p></div><div class="q-explanation"><p>The transformation is</p><ul><li>Horizontal stretch scale factor 0.5</li><li>Translation of <span class="math-tex">\(\left( \begin{matrix} -1 \\ 0 \end{matrix} \right) \)</span></li><li>Reflection in x axis</li><li>Vertical stretch scale factor 3</li><li>Translation of <span class="math-tex">\(\left( \begin{matrix} 0 \\ 1 \end{matrix} \right) \)</span></li></ul></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator">Here is a quiz for the transformation <strong>f(ax + b)</strong> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#3568fd02"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="3568fd02"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Transforming functions f(ax+b) <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-473-875" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) is plotted.</p><p>To draw the graph of <span class="math-tex">\(y=f(\frac{1}{3}x+3)\)</span>, we should translate f(x) by <span class="math-tex">\(\begin{pmatrix} c \\ 0 \end{pmatrix}\)</span> followed by a stretch scale factor <strong><em>d</em></strong> from the y axis.</p><p>Find <strong><em>c</em></strong> and <strong><em>d</em></strong></p></div><div class="q-answer"><p><em><strong>c</strong></em> = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p><p><strong><em>d</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>f(ax + b) is a translation <span class="math-tex">\(\begin{pmatrix} -b \\ 0 \end{pmatrix}\)</span> followed by a stretch scale factor <span class="math-tex">\(\frac{1}{a}\)</span> from the y axis</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) is plotted.</p><p>To draw the graph of <span class="math-tex">\(y=f(\frac{1}{2}(x-3))\)</span>, we should stretch scale factor <strong><em>c</em></strong> from the y axis followed by a translation <span class="math-tex">\(\begin{pmatrix} d \\ 0 \end{pmatrix}\)</span></p><p>Find <strong><em>c</em></strong> and <strong><em>d</em></strong></p></div><div class="q-answer"><p><strong><em>c</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p><strong><em>d</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(y=f(a(x+b))\)</span>is a stretch scale factor <span class="math-tex">\(\frac{1}{a}\)</span> from the y axis followed by a translation <span class="math-tex">\(\begin{pmatrix} -b \\ 0 \end{pmatrix}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = x² is translated 2 units to the left followed by a stretch scale factor 0.5 from the y axis.</p><p>What is the equation of the graph produced?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> y = (2x + 2)²</label></p><p><label class="radio"><input type="radio"> y = (0.5x + 2)²</label></p><p><label class="radio"><input type="radio"> y = (0.5x + 1)²</label></p><p><label class="radio"><input type="radio"> y = (2x + 1)²</label></p></div><div class="q-explanation"><p>f(2x + 2) is a translation <span class="math-tex">\(\begin{pmatrix} -2 \\ 0 \end{pmatrix}\)</span> followed by a stretch scale factor <span class="math-tex">\(\frac{1}{2}\)</span> from the y axis</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of the function f(x) = lnx is shown.</p><p>The function is translated by <span class="math-tex">\(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\)</span> then stretched by scale factor 2 from the y axis.</p><p>What is the new function?</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl3/q3.bmp" style="width: 400px; height: 270px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> ln0.5(x - 2)</label></p><p><label class="radio"><input class="c" type="radio"> ln(0.5x - 2)</label></p><p><label class="radio"><input type="radio"> ln(0.5x + 2)</label></p><p><label class="radio"><input type="radio"> ln(2x - 2)</label></p></div><div class="q-explanation"><p>f(ax + b) is a translation <span class="math-tex">\(\begin{pmatrix} -b \\ 0 \end{pmatrix}\)</span> followed by a stretch scale factor <span class="math-tex">\(\frac{1}{a}\)</span> from the y axis</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = x² is translated <span class="math-tex">\(\begin{pmatrix} -1 \\ 0 \end{pmatrix}\)</span> then stretched by a scale factor <span class="math-tex">\(\frac{1}{3}\)</span> from the y axis</p><p>Which of the following describes the equation of the new graph</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(y=(\frac{1}{3}x+1)^2\)</span></label></p><p><label class="radio"><input type="radio"> y = 3(x + 1)²</label></p><p><label class="radio"><input type="radio"> y = 3x² - 1</label></p><p><label class="radio"><input class="c" type="radio"> y = (3x + 1)²</label></p></div><div class="q-explanation"><p>f(ax + b) is a translation <span class="math-tex">\(\begin{pmatrix} -b \\ 0 \end{pmatrix}\)</span> followed by a stretch scale factor <span class="math-tex">\(\frac{1}{a}\)</span> from the y axis</p><p>Therefore</p><p>y = (3x + 1)² is a translation <span class="math-tex">\(\begin{pmatrix} -1 \\ 0 \end{pmatrix}\)</span> followed by a stretch scale factor <span class="math-tex">\(\frac{1}{3}\)</span> from the y axis</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) is shown in blue</p><p>If g(x) = f(0.5x - 2) which of the following shows the correct graph of y = g(x)</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl3/5a.bmp" style="width: 300px; height: 203px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl3/5b.bmp" style="width: 300px; height: 203px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl3/5c.bmp" style="width: 300px; height: 203px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl3/5d.bmp" style="width: 300px; height: 203px;"></p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> D</label></p></div><div class="q-explanation"><p>To get the graph g(x) = f(0.5x - 2) we start with f(x) then translate <span class="math-tex">\(\begin{pmatrix}2\\ 0 \end{pmatrix}\)</span> then stretch by scale factor 2 from the y axis. The maximum point goes from (0 , 4) to (2 , 4) to (4 , 4). Hence C.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) is shown in blue</p><p>If g(x) = f(-2x + 1) which of the following shows the correct graph of y = g(x)</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl3/6a.bmp" style="width: 300px; height: 203px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl3/6b.bmp" style="width: 300px; height: 203px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl3/6c.bmp" style="width: 300px; height: 203px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl3/6d.bmp" style="width: 300px; height: 203px;"></p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> D</label></p></div><div class="q-explanation"><p>To get the graph g(x) = f(-2x + 1) we start with f(x) then translate <span class="math-tex">\(\begin{pmatrix} -1\\ 0 \end{pmatrix}\)</span> then stretch by scale factor 0.5 from the y axis and a reflection in the y axis. Point A goes from (2 , 3) to (1 , 3) to (0.5 , 3) to (-0.5 , 3). Hence D.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) and y = (ax + b) are plotted below.</p><p>What are the values of <em><strong>a </strong></em>and <em><strong>b</strong></em> ?</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl3/q7.bmp" style="width: 400px; height: 270px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>f(ax + b) is a translation <span class="math-tex">\(\begin{pmatrix} -b \\ 0 \end{pmatrix}\)</span> followed by a stretch scale factor <span class="math-tex">\(\frac{1}{a}\)</span> from the y axis</p><p>f(-0.5x + 1) is a translation <span class="math-tex">\(\begin{pmatrix} -1 \\ 0 \end{pmatrix}\)</span> followed by a stretch scale factor 2 from the y axis and a reflection in the y axis</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Two transformations are performed on a function:</p><p>A graph is translated 9 units right followed by a stretch scale factor <span class="math-tex">\(\frac{1}{3}\)</span> from the y axis.</p></div><div class="q-answer"><div class="q-question"><p>This is the same as a stretch scale factor <span class="math-tex">\(\frac{1}{3}\)</span> from the y axis followed by a translation <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> units right.</p></div></div><div class="q-explanation"><p><strong>f(3(x - 3)) = f(3x - 9)</strong></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) and y = (ax + b) are plotted below.</p><p>What are the values of <em><strong>a </strong></em>and <em><strong>b</strong></em> ?</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl3/q10.bmp" style="width: 400px; height: 311px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p></div><div class="q-explanation"><p>f(ax + b) is a translation <span class="math-tex">\(\begin{pmatrix} -b \\ 0 \end{pmatrix}\)</span> followed by a stretch scale factor <span class="math-tex">\(\frac{1}{a}\)</span> from the y axis</p><p>f(2x - 1) is a translation <span class="math-tex">\(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\)</span> followed by a stretch scale factor <span class="math-tex">\(\frac{1}{2}\)</span> from the y axis</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here's a quiz for the graphs of functions |f(x)| , f|x| , <span class="math-tex">\({ 1\over f}\)</span> and [f(x)]²</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#1691eef5"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="1691eef5"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Transforming Functions HL <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-245-875" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The graph of f(x) is shown in red</p><p>Which of the following below best represents the graph of f(|x|)</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q1_" style="width: 300px; height: 257px;"></p><hr class="hidden-separator"><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q1a" style="width: 300px; height: 292px;"> A</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q1b" style="width: 300px; height: 292px;"> B</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q1c" style="width: 300px; height: 292px;"> C</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q1d" style="width: 300px; height: 292px;"> D</p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> D</label></p></div><div class="q-explanation"><p>Any part of the function <strong><em>f </em></strong> to the right of the y axis is reflected in the y axis</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of f(x) is shown in green</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q2_1" style="width: 350px; height: 309px;"></p><hr class="hidden-separator"><p>Which of the following functions best represents the graph in red</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q2_2" style="width: 350px; height: 310px;"></p><hr class="hidden-separator"></div><div class="q-answer"> <p><label class="radio"><input type="radio"> Neither of these answers</label></p><p><label class="radio"><input type="radio"> f(|x|)</label></p><p><label class="radio"><input class="c" type="radio"> |f(x)|</label></p><p><label class="radio"><input type="radio"> |f(|x|)|</label></p></div><div class="q-explanation"><p><label class="radio">For the transformation |f(x)| , any part of the graph below the x axis is reflected in the x axis.</label></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of f(x) = tanx and <span class="math-tex">\(g(x)=\frac{1}{tanx}\)</span>, for <span class="math-tex">\( (-\frac{\pi}{2}, \frac{\pi}{2} )\)</span><span class="math-tex"><span class="math-tex"><x<\frac{\pi}{2}\)< span=""><x<\frac{\pi}{2}\)< span=""><span class="math-tex"><x<\frac{\pi}{2}\)< span=""><span class="math-tex"><x<\frac{\pi}{2}\)< span="">intersect at the following points</x<\frac{\pi}{2}\)<></span></x<\frac{\pi}{2}\)<></span></x<\frac{\pi}{2}\)<></x<\frac{\pi}{2}\)<></span></span></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\( (\frac{\pi}{4}, 1)\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\( (-\frac{\pi}{4}, -1)\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\( (-\frac{\pi}{4}, 1)\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\((1,1)\)</span></label></p></div><div class="q-explanation"><p>f(x) and <span class="math-tex">\(1\over f(x)\)</span> intersect when <span class="math-tex">\(f(x)=\frac{1}{f(x)}=\pm 1\)</span> </p><p><span class="math-tex">\(tan(\frac{\pi}{4})=1\\ tan(-\frac{\pi}{4})=-1\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of f(x) is shown in red</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q4" style="width: 350px; height: 265px;"></p><hr class="hidden-separator"><p>Which of the following functions best represents the graph in blue</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q4a" style="width: 350px; height: 265px;"></p><hr class="hidden-separator"></div><div class="q-answer"> <p><label class="radio"><input type="radio"> |f(x)|</label></p><p><label class="radio"><input type="radio"> Neither of these answers</label></p><p><label class="radio"><input type="radio"> f(|x|)</label></p><p><label class="radio"><input class="c" type="radio"> |f(|x|)|</label></p></div><div class="q-explanation"><div>any part of the function below the x axis is reflected in the x axis</div><div>and<br>any part of the function to the right of the y axis is reflected in the y axis</div></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>f(x) = cosx</p><p>Which of the following points lie on the graph of <span class="math-tex">\(\frac{1}{f(x)}\)</span></p><p><em><strong>Select all correct answers</strong></em></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> (0 , 1)</label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\((\frac{\pi}{4},\sqrt{2})\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\((-\frac{\pi}{6},2)\)</span></label></p><p><label class="checkbox"><input type="checkbox"> (0 , 0)</label></p></div><div class="q-explanation"><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q5" style="width: 350px; height: 268px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of f(x) is shown in red</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q6" style="width: 350px; height: 243px;"></p><p>Which of the following best represents a sketch of the graph of <span class="math-tex">\(y= {1 \over f(x)}\)</span></p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q6a.png" style="width: 350px; height: 241px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl/q6b.png" style="width: 350px; height: 243px;"></p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q6c.png" style="width: 350px; height: 243px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl/q6d.png" style="width: 350px; height: 246px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>B</span></label> </p><p><label class="radio"> <input type="radio"> <span>C</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>D</span></label> </p><p><label class="radio"> <input type="radio"> <span>A</span></label> </p></div><div class="q-explanation"><p>The x intercepts of y = f(x) become vertical asymptotes</p><p>The graph of y = f(x) and <span class="math-tex">\(y= {1 \over f(x)}\)</span> intersect where <span class="math-tex">\(f(x)= \pm 1\)</span></p><p>Local maxima of f becomes local minima of <span class="math-tex">\(y= {1 \over f(x)}\)</span></p><p>Hence D is the best sketch of <span class="math-tex">\(y= {1 \over f(x)}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of f(x) is shown in red</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q7" style="width: 350px; height: 348px;"></p><p>Which of the following represents the graph of y = [f(x)]²</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q7a" style="width: 350px; height: 348px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl/q7b" style="width: 350px; height: 348px;"></p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q7c" style="width: 350px; height: 348px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl/q7d" style="width: 350px; height: 348px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>A</span></label> </p><p><label class="radio"> <input type="radio"> <span>B</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>D</span></label> </p><p><label class="radio"> <input type="radio"> <span>C</span></label> </p></div><div class="q-explanation"><p>The output of the function is squared.</p><p>As all outputs will be positive, there should be no part of the graph below the x axis</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><p>A</p><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of f(x) = sinx is shown in red</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q8" style="width: 350px; height: 256px;"></p><p>Which of the following represents the graph of y = [f(x)]²</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q8a" style="width: 350px; height: 224px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl/q8b" style="width: 350px; height: 224px;"></p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q8c" style="width: 350px; height: 224px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl/q8d" style="width: 350px; height: 224px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>A</span></label> </p><p><label class="radio"> <input type="radio"> <span>C</span></label> </p><p><label class="radio"> <input type="radio"> <span>D</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>B</span></label> </p></div><div class="q-explanation"><p>The local maxima for f(x) = sinx is at <span class="math-tex">\((\frac{\pi}{4},1)\)</span>.</p><p>Therefore the local maxima for [f(x)]² is at <span class="math-tex">\((\frac{\pi}{4},1^2)\)</span>...the same point.</p><p>The local minima for [f(x)]² is at <span class="math-tex">\((-\frac{\pi}{4},1^2)\)</span>.</p><p>The answer could be B or C.</p><p>C is the graph of |f(x)|, since the shape of the curves is the same as f(x).</p><p>B is the correct answer because the outputs of [f(x)]²<f(x) 0="" for=""> </f(x)></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of f(x) is shown in red</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q9" style="width: 350px; height: 288px;"></p><p>Which of the following represents the graph of y = [f(x)]²</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q9a" style="width: 350px; height: 252px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl/q9b" style="width: 350px; height: 224px;"></p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q9c" style="width: 350px; height: 288px;"> <img alt="" src="../../files/functions/transforming-functions/quizhl/q9d" style="width: 350px; height: 252px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>D</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>B</span></label> </p><p><label class="radio"> <input type="radio"> <span>A</span></label> </p><p><label class="radio"> <input type="radio"> <span>C</span></label> </p></div><div class="q-explanation"><p>All outputs for [f(x)]² are positive so A is definitely wrong<br>f(x) goes through the point (1 , 1)</p><p>Hence, [f(x)]² goes through the point (1² , 1²) ...the same point.</p><p>Hence B is the correct answer.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) is shown below</p><p><img alt="" src="../../files/functions/transforming-functions/quizhl/q10(1)" style="width: 350px; height: 241px;"></p><p>Which of the following is correct?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(|f(-1)|<[f(-1)]²<\frac{1}{f(1)}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{1}{f(1)}<[f(-1)]²<|f(-1)|\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\([f(-1)]²<|f(-1)|<\frac{1}{f(1)}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{1}{f(1)}<|f(-1)|<[f(-1)]²\)</span></span></label> </p></div><div class="q-explanation"><p>f(-1) < 1</p><p>Therefore, [f(-1)]² < |f(-1)|</p><p>And f(-1)]² < |f(-1)| < 1</p><hr class="hidden-separator"><p>f(1) < 1</p><p>Therefore <span class="math-tex">\(\frac{1}{f(1)}>1\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="540"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>a) The following diagram shows the graph of a function <strong><em>f</em></strong></p> <p>On the same set of axes, sketch the graph of <strong><em>f(-x) + 2</em></strong></p> <p><strong><em>You can print this graph from <a href="../../files/functions/transforming-functions/esq1a.pdf">here</a></em></strong></p> </div> <p><img alt="" src="../../files/functions/transforming-functions/esq1a.png" style="width: 414px; height: 417px;"></p> <p>b)</p> <div> <p>The following diagram shows the function <strong><em>af(x+b)</em></strong></p> <p>Write down the values<strong><em> of a </em></strong>and<strong><em> b </em></strong></p> <p><img alt="" src="../../files/functions/transforming-functions/esq1b.png" style="width: 414px; height: 417px;"></p> </div> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">a) Think about which two transformations are represented by <strong><em>f(-x) + 2</em></strong> and apply them to the 4 points.<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/transforming-functions/esq_transformingfunctions1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/transforming-functions/esq_transformingfunctions1.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="541"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The graph of f(x) has a local maxima at <span class="math-tex">\((1 - a , 2b)\)</span> and a local minima at <span class="math-tex">\((3a,b-3)\)</span>.</p> <p>a) Find the coordinates of the local maxima of <span class="math-tex">\(f(x+a)-2b\)</span></p> <p>b) Find the coordinates of the local minima of <span class="math-tex">\(2f(3x)\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>It helps to sketch a graph of what the function might look like then mark on the local maxima and minima</p> <p>a) Consider the 2 transformations for <span class="math-tex">\(f(x+a)-2b\)</span> and apply them to the point <span class="math-tex">\((1 - a , 2b)\)</span><br> b) Consider the 2 transformations for <span class="math-tex">\(2f(3x)\)</span>and apply them to the point <span class="math-tex">\((3a,b-3)\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/transforming-functions/esq_transformingfunctions2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/transforming-functions/esq_transformingfunctions2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="544"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Consider the function <strong><em>f(x) = x<sup>3</sup> - 4x² - x + 6 </em></strong>, <span class="math-tex">\(x \in \mathbb{R}\)</span></p> <p>The graph of <strong><em>f</em></strong> is translated two units to the left and 3 units up to form the function <strong><em>g(x)</em></strong>. Express <strong><em>g(x)</em></strong> in the form <strong><em>ax<sup>3 </sup>+ bx² + cx + d </em></strong> where <span class="math-tex">\(a,b,c,d \in \mathbb{Z}\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><strong><em>g(x) = f(x + 2) + 3</em></strong><content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/transforming-functions/esq_transformingfunctions2b.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/transforming-functions/esq_transformingfunctions2b.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="542"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>The graph of <span class="math-tex">\(y=e^{2x-1}\)</span> is obtained by performing two transformations to the function <span class="math-tex">\(f(x)=e^x\)</span></p> <p>- a stretch of scale factor <strong><em>a</em></strong> parallel to the x axis</p> <p>- a stretch of scale factor <strong><em>b </em></strong>parallel to the y axis.</p> <p>Find the values of <strong><em>a</em></strong> and <strong><em>b</em></strong></p> </div> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Use the properties of indices to re-write the equation of the graph</p> <p><span class="math-tex">\(m^x\times m^y=m^{x+y}\)</span></p> <content> </content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/transforming-functions/esq_transformingfunctions3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/transforming-functions/esq_transformingfunctions3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="543"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Consider the function <strong><em>f(x) = 2x² - 5x + 3</em></strong></p> <p>a) Factorise <strong><em>f(x)</em></strong></p> <p>b) Express <strong><em>f(x)</em></strong> in the form <strong><em>a(x - h)² + k</em></strong></p> <p>c) Hence, sketch the graph of <span class="math-tex">\(\frac{1}{f(x)}\)</span> indicating the equations of the asymptotes, the coordinates of any stationary points and the y intercept.</p> <p>d) Sketch the graph of <span class="math-tex">\(\frac{1}{f|x|}\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>c) Use the previous two parts to help out with this sketch</p> <ul> <li><content></content>part a) helps you find the x intercepts of <strong><em>f</em></strong></li> <li>part b) helps you find the vertex of <strong><em>f</em></strong></li> </ul> <p>Ensure that you give all the details in the graph that are requested.</p> <p>d) You need to use the previous part to sketch this graph - the negative x values will 'behave' like their positive counterparts (reflect the right-hand part of the graph of <span class="math-tex">\(\frac{1}{f(x)}\)</span> in the y axis).</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/transforming-functions/esq_transformingfunctions4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/transforming-functions/esq_transformingfunctions4.pdf" width="640"></iframe></p> </section> </div> </div> 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