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<div id="main-column" class="span9"> <article id="transformations-of-graphs" style="margin-top: 16px;">
<h1 class="section-title">Transformations of graphs</h1>
<ul class="breadcrumb"><li><a title="Home" href="../../../index.html"><i class="fa fa-home"></i></a><span class="divider">/</span></li><li><span class="gray">2. Functions</span><span class="divider">/</span></li><li><span class="active">Transformations of graphs</span></li></ul>
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<p><a class="scroll-to" data-target="TM">teaching materials</a> ↴</p><div class="yellowBg"><p><u>syllabus content</u>: </p><p><strong>SL 2.11</strong> Transformations of graphs; translations, reflections (in both axes), vertical stretches, horizontal stretches, composite transformations</p></div><h4>Teacher notes</h4><p><img alt="" src="../../../ib/mathanalysis/analysis/functions/transformations/trans_img_1-1.jpg" style="float: right; width: 250px; height: 140px;">SL and HL students need to understand the following types of transformations that can be applied to the graph of a function:<br>▪ <strong>translations </strong>- horizontal & vertical<br>▪ <strong>reflections </strong>- in both <em>x</em>- and y<em>-</em>axes<br>▪ <strong>dilations </strong>(stretching & shrinking) - in both horizontal and vertical directions</p><p>Transformations of graphs is a topic/skil that will be taught early in the course during a unit on functions and then will appear again during other times in the course - especially when studying the graphs of trigonometric functions.</p><p>Consider teaching transformations of graphs by having your students individually work through an activity using graphing software - either on their GDC or with computer software. It's best if there is some dynamic aspects to the graphing utility - for example, creating sliders that will change a particular parameter in a function that then simultaneously changes the graph of that function. Below are two images from a 7-page activity that I wrote for the <strong>TI-Nspire</strong> handheld device (GDC) that uses dynamic sliders. Click on either of the images to obtain a PDF copy of the activity that is also listed in the<a class="scroll-to" data-target="TM"> teaching materials</a> section below.</p><p><a href="/media/ib/mathhlsl/files/Teaching%20Materials/Functions-Equations/transformations-graphs/ACT_2-3-30v3_SLHL_transformations_graphs_Nspire.pdf" target="_blank"><img alt="" src="../../../ib/mathanalysis/analysis/functions/transformations/transform_act_img1.jpg" style="width: 300px; height: 225px;"><img alt="" src="../../../ib/mathanalysis/analysis/functions/transformations/transform_act_img2.jpg" style="width: 300px; height: 225px;"></a></p><p>Give your student some interactive practice on transformations of graphs, by assigning the page <u><a href="../43466/transformations-interactive-practice.html" target="_blank"><span style="color:#0000FF;">Transformations - interactive practice</span></a></u> that contains two dynamic Geogebra applets. The first applet lets a student enter a function (the default function is a quadratic) and then to transform the graph of the function by changing the values of four different parameters. The second applet is a dynamic quiz that automatically checks responses.</p><p>Exam questions may include <strong>composite transformations</strong> where there is a sequence of more than transformation to be performed on a function. Sometimes the order in which the transformations are performed makes a difference and sometimes the order does not make a difference. I've made a one-page document with an example illustrating a situation where it does make a difference. Open the following document: <u><a href="/media/ib/mathanalysis/analysis/functions/transformations/sequence-of-transformations-of-a-graph.pdf" target="_blank"><span style="color:#0000FF;">sequence of tranformations of a graph</span></a></u>.</p><h5>Exam-style question on transformations of graphs</h5><p>On the <a href="../32352/sample-mock-exams.html" target="_blank">sample mock exam</a> <strong>MockA_SL_Paper_1_2014_v1</strong>, question 10 part B involved transformations of graphs. The last part of the question (sub-part c) is tricky for SL students (and HL). To see the question and some explanatory notes on sub-parts b and c, open the following document: <u><a href="/media/ib/mathanalysis/analysis/functions/transformations/mocka_sl_p1_2014_q10_part_b.pdf" target="_blank"><span style="color:#0000FF;">MockA_SL_P1_2014_Q10_Part_B</span></a></u>. Also see the notes on <u><a href="/media/ib/mathanalysis/analysis/functions/transformations/sequence-of-transformations-of-a-graph.pdf" target="_blank"><span style="color:#0000FF;">sequence of transformations of a graph</span></a></u> (presented above) that has an example illustrating some guidelines on the sequential order of transformations in a composite transformation.</p><div class="blueBg"><h5>syllabus content HL 2.16</h5><p>HL students also need to be familiar with the relationship between the graph of <span class="math-tex">\(y = f\left( x \right)\)</span> and the graphs of <span class="math-tex">\(y = \left| {f\left( x \right)} \right|\)</span>, <span class="math-tex">\(y = f\left( {\left| x \right|} \right)\)</span>, <span class="tib-mathml"><math><mrow><mi>y</mi><mo>=</mo><mfrac> <mn>1</mn> <mrow> <mi>f</mi><mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow> </mrow></mfrac></mrow></math></span> , <span class="math-tex">\(y = f\left( {ax + b} \right)\)</span> and <span class="math-tex">\(y = {\left[ {f\left( x \right)} \right]^2}\)</span>. See 'curve sketching' below - and also see the <strong>Geogebra applet </strong>on the page <u><a href="../34036/other-changes-to-a-graph.html" target="_blank"><span style="color:#0000FF;">Other changes to a graph</span></a></u>.</p></div><p>Additionally, I firmly believe that the topic which is most suitable for teaching by means of activities that students carry out themselves (usually in the classroom) is the topic of <strong>transformations of graphs</strong>. My experience has been that students can gain a very clear understanding of different graph transformations by working through an activity using graphing software (either on a GDC or a computer) that presents the material in a dynamic and interactive manner. In fact, I find that it takes significantly less time to teach the topic using individual technology-based activities with students compared to a traditional lecture along with static information presented in a paper textbook.</p><h5>Curve sketching (not <em>really </em>transformations of graphs) - mostly for HL students</h5><p>While teaching transformations of graphs (translations, reflections & dilations), I think it's a good idea to give students experience with other types of graph 'changes' (see <u><a href="../34036/other-changes-to-a-graph.html" target="_blank"><span style="color:#0000CD;">Other changes to a graph</span></a></u>). I put this in the general category of <em><strong>curve sketching</strong></em>. For example, when students are studying differential calculus they should be able to sketch the graph of a function given the graph of the derivative of the function (and a point or two to 'anchor' it). Before that students should be exposed to questions like the one below (click on the 'eye'). 'Curve sketching' questions appear more often on HL exams than on SL exams. Nevertheless, questions like these require good conceptual understanding and a willingness to think logically - good for <strong>all</strong> students to learn and practice. Most students find these kinds of questions challenging - where there is no computation involved, just logic and understanding. A student should not be allowed to use a GDC when performing 'curve sketching' questions. Students should be prepared to encounter such questions on the Paper 1 exam.</p><div class="greyBg"><p><strong>'Curve sketching' question</strong></p><p>download worksheet with this question and answers: <a href="Browse server to select file" target="_blank" title="Exercise set"><img class="ico" src="../../../img/icons/exercise-set.png"> AA_HL_changes_to_a_graph</a></p><section class="tib-hiddenbox"><p><img alt="" src="../../../ib/mathanalysis/analysis/functions/transformations/curve_sketch_q1_img.jpg" style="width: 616px; height: 256px;"></p><section class="tib-hiddenbox"><p><img alt="" src="../../../ib/mathanalysis/analysis/functions/transformations/curve_sketch_q1a_img.jpg" style="width: 616px; height: 381px;"></p><section class="tib-hiddenbox"><p><img alt="" src="../../../ib/mathanalysis/analysis/functions/transformations/curve_sketch_q1b_img.jpg" style="width: 616px; height: 359px;"></p><p><a href="/media/ib/mathanalysis/analysis/functions/transformations/curve_sketch_solution.pdf" target="_blank"><span style="color:#0000FF;"><strong>solution</strong></span></a></p></section></section></section></div><h3><strong><u>4 questions</u> - </strong>‘accessible’ to ‘discriminating’</h3><p>download: <a href="/media/ib/mathhlsl/files/Teaching%20Materials/Algebra/seq-series/4_Qs_sequences_series_1_solutions_notes.pdf" target="_blank" title="Exercise set"> </a><a href="/media/ib/mathhlsl/images/_Site/Algebra/exponents%20%26%20logarithms/4_Qs_exponents_logarithms_1_with_answers.pdf" target="_blank" title="Exercise set"> </a><a href="/media/ib/mathanalysis/analysis/functions/transformations/4_qs_transformations_graphs_1_with_answers.pdf" target="_blank" title="Exercise set"><img class="ico" src="../../../img/icons/exercise-set.png"> 4_Qs_transformations_graphs_1_with_answers</a> <img alt="" class="ico" height="16" src="../../../img/tib-icons/standard-level-16.png" title="" width="16"> <img alt="" class="ico" height="16" src="../../../img/tib-icons/higher-level-16.png" title="" width="16"></p><div class="pinkBg"><p><strong>accessible SL question</strong></p><section class="tib-hiddenbox"><p><img alt="" src="../../../ib/mathanalysis/analysis/functions/transformations/q1_transform.jpg" style="width: 585px; height: 113px;"></p></section></div><div class="blueBg"><p><strong>moderate SL / accessible HL question</strong></p><section class="tib-hiddenbox"><p><img alt="" src="../../../ib/mathanalysis/analysis/functions/transformations/q2_transform.jpg" style="width: 603px; height: 103px;"></p></section></div><div class="yellowBg"><p><strong>discriminating SL / moderate HL question</strong></p><section class="tib-hiddenbox"><p><img alt="" src="../../../ib/mathanalysis/analysis/functions/transformations/q3_transform.jpg" style="width: 598px; height: 355px;"></p></section></div><div class="greenBg"><p><strong>discriminating HL question</strong></p><section class="tib-hiddenbox"><p><img alt="" src="../../../ib/mathanalysis/analysis/functions/transformations/q4_transform.jpg" style="width: 528px; height: 72px;"></p></section></div><div class="yellowBg"><p><strong>Answers</strong></p><section class="tib-hiddenbox"><p><img alt="" src="../../../ib/mathanalysis/analysis/functions/transformations/ans2_transform.jpg" style="width: 600px; height: 559px;"></p></section></div><p><strong><a class="anchor" id="TM" name="TM"> </a>♦ teaching materials</strong></p><p><a href="/media/ib/mathanalysis/analysis/functions/transformations/aa_slhl_transformations_graphs.pdf" target="_blank" title="Exercise set"><img class="ico" src="../../../img/icons/exercise-set.png"> AA_SLHL_transformations_graphs</a> <img alt="" class="ico" height="16" src="../../../img/tib-icons/standard-level-16.png" title="" width="16"> <img alt="" class="ico" height="16" src="../../../img/tib-icons/higher-level-16.png" title="" width="16"><br>A set of 7 questions covering the full range of transformations of graphs - translations, reflections & dilations (<strong>answers </strong>included)</p><p><a href="/media/ib/mathanalysis/analysis/functions/transformations/transformations_graphs_nspire.pdf" target="_blank" title="Activities"><img class="ico" src="../../../img/icons/activities.png"> transformations_graphs_Nspire</a> <img alt="" class="ico" height="16" src="../../../img/tib-icons/standard-level-16.png" title="" width="16"> <img alt="" class="ico" height="16" src="../../../img/tib-icons/higher-level-16.png" title="" width="16"><br>An activity that users sliders and dynamic graphs to teach transformations of graphs; students will answer a series of questions/conjectures based on their observations; suitable for beginners to the TI-Nspire - contains thorough instructions supported by images (7 pages)</p><script>document.querySelectorAll('.tib-teacher-only').forEach(e => e.remove());</script>
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