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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 SL</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. 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"> <div> <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> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <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"> <p>These graphs should help you understand the transformations of functions</p> <div class="panel panel-yellow panel-has-colored-body 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><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-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"> <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="../1801/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-yellow 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>Translations y = f(x - a) + b</p> </div> </div> <div class="panel-body"> <p>Click on the image to open an applet and explore this transformation: </p> <div> <p><a href="../1802/translations-.html" target="_blank"><img alt="" src="../../files/functions/transforming-functions/translation-thumbnail.png" style="width: 300px; height: 340px;"></a></p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </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_functions.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/functions/transforming-functions/revision-notes_transforming_functions.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"> <p>Here is a quiz that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#06b1b20d"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="06b1b20d"> <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="11-217-677" 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> </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-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 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" 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