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class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1061/functions.html">Functions</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">2.3 & 2.4 Key Features of Functions And Graphing </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"> <div class="intro-card"> <div class="bg-cover" style="background-image: url('/media/mathsapplications/images/pexels-karyme-franca-1458036.jpg');"> </div> <img src="../../images/pexels-karyme-franca-1458036-1.jpg" style="display: none"> <div class="content"> <h4 class="theading">How might we analyse a function?</h4> <p class="text">Now that we know something about the basics of functions, this section aims to help deepen your understanding of them and enable you to identify key features such as vertex points, intercepts with axes and asymptotes. You will also learn about the difference between drawing and sketching a graph, alongside finding points were graphs intersect. </p> </div> </div> <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><span id="docs-internal-guid-2de417ec-7fff-cd31-daa2-b7d0dfa8a7cc"></span></p> <p dir="ltr"><span id="docs-internal-guid-2de417ec-7fff-cd31-daa2-b7d0dfa8a7cc">In this unit you should learn to…</span></p> <ul style="list-style-type:disc;"> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Identify the key features of graphs, such as its vertices, asymptotes, y-intercepts and zeroes. Specifically for quadratics, this may also include the axis of symmetry. </p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Know how to draw a graph and sketch a graph. </p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Know how to solve equations based on graphing both sides and finding the point of intersection.</p> </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>The following is a series of videos that will help you understand, learn about and review this sub-topic.</p> <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>1. Key Features of Graphs </p> </div> </div> <div class="panel-body"> <div> <p>This video looks at the most significant features of graphs.<iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/454498102" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. Key Features: Exam Questions</p> </div> </div> <div class="panel-body"> <div> <p>A few exam style questions based on finding key features of graphs. <iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/454499140" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Drawing Graphs</p> </div> </div> <div class="panel-body"> <div> <p>What does it mean to "Draw a graph" and how does one achieve maximum marks in an exam. <iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/454989684" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>4. Sketching Graphs</p> </div> </div> <div class="panel-body"> <div> <p>What's the difference between drawing and sketching a graph? <iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/454500522" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>5. Intersection points of Curves</p> </div> </div> <div class="panel-body"> <div> <p>This quick video looks at solving equations graphically, which essentially boils down to finding points of intersection.<iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/454501019" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div id="carousel-268" class="dynamic-gallery carousel slide" data-id="268"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-268/flashcard1.png" data-fancybox="gallery-268" title="" data-caption=""><img alt="" src="../../../std-galleries/15-268/flashcard1.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-268/flashcard2.png" data-fancybox="gallery-268" title="" data-caption=""><img alt="" src="../../../std-galleries/15-268/flashcard2.png"></a></div></div><a class="left carousel-control" href="#carousel-268" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-268" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-268/flashcard1-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-268/flashcard2-thumb128.jpg"><li></li></ol> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test yourself</p> </div> </div> <div class="panel-body"> <div> <h4>Self Checking Quiz</h4> <p>Practice your understanding on these quiz questions. Check your answers when you are done and read the worked solutions when you get stuck. If you find there are still some gaps in your understanding then go back to the videos and above<em>.</em></p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="tib-quiz" data-stats="15-554-2127"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>What can you call the points shown on the diagram below?</p><p><img alt="" src="../../picture1(9).png" style="width: 632px; height: 490px;"></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> Zeros</label></p><p><label class="checkbox"><input class="c" type="checkbox"> Roots</label><label class="radio"></label></p><p><label class="checkbox"><input class="c" type="checkbox"> x-intercepts</label></p><p><label class="checkbox"><input type="checkbox"> Vertices</label></p></div><div class="q-explanation"><p>The only incorrect response to this question is "vertices". </p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>A function, <span class="math-tex">\(f\)</span>, is defined by <span class="math-tex">\(f\left(x\right)=x^{2}+6x+3\)</span></p><p>Find the x-intercepts of the graphs of <span class="math-tex">\(f\)</span></p></div><div class="q-answer"><p>Answers: <input type="text" style="height: auto;" data-c="-5.45"> <span class="review"></span> , <input type="text" style="height: auto;" data-c="-0.551"> <span class="review"></span> </p></div><div class="q-explanation"><p>Notice that the answers have not been given as co-ordinates. This is strictly correct, but not necessary in an exam scenario. Answers should also be given to 3 s.f. where appropriate. </p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of these function contain a vertical asymptote? (there is more than one correct answer)</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span><span class="math-tex">\(f\left(x\right)=\frac{1}{x}\)</span></span></label> </p><p><label class="checkbox"><input class="c" type="checkbox"> <span><span class="math-tex">\(f\left(x\right)=\frac{6x+2}{x-3}\)</span></span></label> </p><div class="ck-mark" id="right-mark" style="display:none;"> </div><p><label class="checkbox"><input type="checkbox"> <span><span class="math-tex">\(f\left(x\right)=3^{x}-2\)</span></span></label> </p><p><label class="checkbox"><input type="checkbox"> <span><span class="math-tex">\(f\left(x\right)=x^{3}+3x^{2}-2x\)</span></span></label> </p><p><label class="checkbox"><input type="checkbox"> <span><span class="math-tex">\(f\left(x\right)=\sin\left(x\right)\)</span></span></label> </p><div class="ck-mark" id="wrong-mark" style="display:none;"> </div></div><div class="q-explanation"><p>Only the first two rational functions contain vertical asymptotes (they also contain horizontal asymptotes). The exponential function, <span class="math-tex">\(f\left(x\right)=3^{x}-2\)</span>, contains a horizontal asymptote but no vertical asymptote. </p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Tick the answers below which are most commonly required when you are asked to "sketch a graph."</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span>Key features of the graph, such as x-intercepts and y-intercepts, should be clearly labelled. </span></label> </p><p><label class="checkbox"><input class="c" type="checkbox"> <span>Axes must be labelled.</span></label> </p><div class="ck-mark" id="right-mark" style="display:none;"> </div><p><label class="checkbox"><input type="checkbox"> <span>You must you graph paper.</span></label> </p><p><label class="checkbox"><input type="checkbox"> <span>You must label every number on each axis. </span></label> </p><p><label class="checkbox"><input type="checkbox"> <span>You must use a pencil.</span></label> </p></div><div class="q-explanation"><p>A sketch requires you to be relatively accurate, but it does not require the same level of detail as when you are asked to "draw a function."</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Tick the answers below which are most commonly required when you are asked to "draw a graph."</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> You must use a pencil and a ruler.</label></p><p><label class="checkbox"><input class="c" type="checkbox"> You must adhere to any scales given for the axes. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> You must use graph paper.</label></p><p><label class="checkbox"><input class="c" type="checkbox"> The curve must pass directly through any points plotted. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> You must plot individual points on the graph if you are given the data to do so. </label></p></div><div class="q-explanation"><p>Drawing a graph requires precision, hence why all answers above are correct. </p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>The formuala, <span class="math-tex">\(A=\pi{r}^{2}+\frac{100}{r}\)</span>, is given for the surface area, <span class="math-tex">\(A\)</span>, of a cylinder with radius, <span class="math-tex">\(r\)</span>.</p><p>Find the local minimum point which minimises the surface area. </p></div><div class="q-answer"><p>Minimum Surface Area: <input type="text" style="height: auto;" data-c="59.6"> <span class="review"></span> <span class="math-tex">\({cm}^2\)</span></p></div><div class="q-explanation"><p>Graph the function in your GDC and identify the local minimum. On the GDC you will be given (2.52, 59.6), but you can discard 2.52 given that this is the value of the radius, <span class="math-tex">\(r\)</span>, which results in a minimum surface area, and that is not needed for the question.</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Below is a sketch of the function, <span class="math-tex">\(f\left(x\right)=\frac{1}{3}x^{3}+\frac{3}{4}x^{2}-x-1\)</span></p><p>Identify what is missing from the skecth below:</p><p><img alt="" src="../../picture2(5).png" style="width: 686px; height: 480px;"></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> Vertex points have not been labelled.</label></p><p><label class="checkbox"><input type="checkbox"> The sketch should include numbers along both axes.</label></p><p><label class="checkbox"><input class="c" type="checkbox"> The x-intercepts and y-intercept has not been labelled. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> The axes have not been labelled. </label></p></div><div class="q-explanation"><p>See answers given.</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function, <span class="math-tex">\(f\left(x\right)=3^{x}+\frac{1}{x+2}\)</span> </p><p>Sketch the grap of <span class="math-tex">\(y=f(x)\)</span> for <span class="math-tex">\({-2}\leq{x}\leq{2}\)</span> and <span class="math-tex">\({-2}\leq{y}\leq{10}\)</span>.</p></div><div class="q-answer"><p>Type "done" into the textbook to check your answer: <input type="text" style="height: auto;" data-c="done"> <span class="review"></span> </p></div><div class="q-explanation"><p><img alt="" src="../../picture1(10).png" style="width: 701px; height: 448px;"></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the functions <span class="math-tex">\(f\left(x\right)=x+2\)</span> and <span class="math-tex">\(g\left(x\right)=3^{x}-3\)</span></p><p>Solve <span class="math-tex">\(f\left(x\right)=g\left(x\right)\)</span>.</p></div><div class="q-answer"><p>Answers: <input type="text" style="height: auto;" data-c="-5.00"> <span class="review"></span> , <input type="text" style="height: auto;" data-c="1.74"> <span class="review"></span> </p></div><div class="q-explanation"><p>When you graph the two functions, there are two points of intersection, (-5.00, -3.00) and (1.74, 3.74).</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Finish the sentence - </p><p>A straight line can intersect a quadratic function in:</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> 2 points</label></p><p><label class="checkbox"><input class="c" type="checkbox"> No points </label></p><p><label class="checkbox"><input class="c" type="checkbox"> 1 point - if it is a tangent to the curve. </label></p></div><div class="q-explanation"><p>See answers.</p></div><div class="actions"><span class="score" data-score="0"></span><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="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </section> </div> </div> </div> <h4> </h4> <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"> <p>The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.</p> <div class="panel panel-default panel-has-colored-body"> <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> <p><img alt="" src="../../eq1.png" style="width: 625px; height: 201px;"></p> <p>Video solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/456919251"></iframe></div> </section> </div> </div> </div> <div class="panel panel-has-colored-body panel-default 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> <p><img alt="" src="../../files/eq2.png" style="width: 625px; height: 133px;"></p> <p>Video solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/456938128" webkitallowfullscreen="" width="100%"></iframe></section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default 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> <p><img alt="" src="../../files/picture1.png" style="width: 625px; height: 158px;"></p> <p>Video solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/457518665" webkitallowfullscreen="" width="100%"></iframe></section> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2127"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>2.3 & 2.4 Key Features of Functions And Graphing </strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your notes here..."></textarea> </div> <div class="panel-footer text-xs-center"> <span id="last-edited" class="mb-xs-3"> </span> <div class="actions mt-xs-3"> 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