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function</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../1804/translations.html">Translations</a></label></li></ul></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../681/factor-and-remainder-theorem.html">Factor and Remainder Theorem</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../680/sums-and-products-of-roots.html">Sums and Products of Roots</a></label></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> Quadratics <a href="#" class="mark-page-favorite pull-right" data-pid="679" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol 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">Quadratics</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/quadratics/main-1.jpg" style="float: left; width: 100px; height: 100px;">On this page we will look at quadratic expressions and how the different factorised forms link to the shape of the graph. We will also get an understanding about how the discriminant affects not only the number of roots of a quadratic equation, but also how we can use it as a tool to help us solve problems with intersections of graphs.</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 to</p> <ul> <li>Understand the quadratic function <span class="math-tex">\(f(x)=ax^2+bx+c\)</span> and its graph</li> <li>Find and use the intercept form <span class="math-tex">\(f(x)=a(x-p)(x-q)\)</span></li> <li>Find and use the vertex form <span class="math-tex">\(f(x)=a(x-h)^2+k\)</span></li> <li>Solve quadratic equations</li> <li>Solve quadratic inequalities</li> <li>Use the discriminant <span class="math-tex">\(\Delta =b^2-4ac\)</span> to determine the nature of roots</li> </ul> </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"> <div> <p>The following videos will help you understand all the concepts from this page</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>Quadratics and Graphs</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="502"> <p>In the following video, we look at quadratic expressions and their different factorised forms, how we might solve quadratic equations and <strong>most importantly </strong>how this all links to the graphs of the quadratic function.</p> <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/308684661"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</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/quadratics/quadratics-and-graphs.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/quadratics-and-graphs.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-yellow panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Quadratics and the Discriminant</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="503"> <p>In the following video, we are going to look at quadratic equations and the discriminant. For the general quadratic equation ax² + bx + c = 0, the discriminant is given by <span class="math-tex">\(\triangle ={ b }^{ 2 }-4ac\)</span></p> <p>We will see that if</p> <ul> <li><strong>b² – 4ac > 0 , there are 2 distinct real roots</strong></li> <li><strong>b² – 4ac = 0 , there is 1 repeated real root</strong></li> <li><strong>b² – 4ac < 0 , there are 0 real roots</strong></li> </ul> <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/309236866"></iframe></div> <p>Here you can play with the applet used in the video. Change the parameters of the quadratic function f(x) = ax² + bx + c and check the number of roots when the discriminant is positive, zero or negative.</p> <p style="text-align: center"><iframe height="508px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/crmuktdm/width/576/height/508/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="Quadratics and the Discriminant" width="576px"></iframe></p> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</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/quadratics/quadratics-and-the-discriminant.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/quadratics-and-the-discriminant.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-yellow panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Discriminant Example</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="507"> <p>In the following video, we will look at a typical exam-style question involving quadratic functions. One of the challenges that we sometimes face with exam questions is that it is not always obvious what method we need to use to solve the problem. Often trying to visualise the situation with a graph helps. It certainly does in the question below:</p> <p>Let f(x) = 2x² + kx + 3</p> <p>Find the values of <strong><em>k</em></strong> for which <em>f(x) </em>> 0 , for all <em>x</em></p> <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/317503421"></iframe></div> <p>If you want to explore the graph in the question, you might find the following applet useful</p> <p style="text-align: center"><iframe height="440px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/s5jtarqn/width/554/height/440/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="Quadratics and discriminant ESQ" width="554px"></iframe></p> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</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/quadratics/quadratics-and-the-discriminant-esq.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/quadratics-and-the-discriminant-esq.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-yellow panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Discriminant and Intersecting Graphs</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="515"> <p>In the following video, we will look at a peculiar application of the discriminant.</p> <p>If a quadratic function f meets the linear function g at a tangent, then the solution to the equation f = g has one (repeated) root. In other words, the f = g gives a quadratic equation and the discriminant = 0</p> <p><img alt="" src="../../files/functions/quadratics/intersecting-graphs.jpg" style="width: 350px; height: 307px;"></p> <p>Here is the question that we will consider to look at this case:</p> <hr class="hidden-separator"> <p><em><strong>The line y = kx + 20 is a tangent to the curve f where f(x) = 12 -2x -2x²</strong></em></p> <p><em><strong>Find the values of k.</strong></em></p> <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/319678367"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</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/quadratics/discriminant-and-intersecting-graphs.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/discriminant-and-intersecting-graphs.pdf" width="640"></iframe></p> </section> </div> </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 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/quadratics/quadratics_revision-notes.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/functions/quadratics/quadratics_revision-notes.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="#73d31645"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="73d31645"> <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%;"> Quadratics and Graphs <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="11-210-679" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Drag the expressions in the correct place to complete the factorising of the following quadratic expressions</p><p> <span class="q-text-draggable draggable" draggable="true">(x - 3)</span> <span class="q-text-draggable draggable" draggable="true">(x + 3)</span> <span class="q-text-draggable draggable" draggable="true">(x + 2)</span> <span class="q-text-draggable draggable" draggable="true">(x - 2)</span> <span class="q-text-draggable draggable" draggable="true">(x - 8)</span></p></div><div class="q-answer"><p>2x² - 10x + 12 = 2 (x - 3)<input type="text" style="height: auto;" data-c="(x - 2)"> <span class="review"></span></p><p>3x² - 3x - 16 = 3 (x + 2)<input type="text" style="height: auto;" data-c="(x - 3)"> <span class="review"></span></p><p>x² - 9 = (x - 3)<input type="text" style="height: auto;" data-c="(x + 3)"> <span class="review"></span></p><p>3x² + 7x - 6 = (3x - 2)<input type="text" style="height: auto;" data-c="(x + 3)"> <span class="review"></span></p><p>x² + 6x + 9 = <input type="text" style="height: auto;" data-c="(x + 3)"> <span class="review"></span> <input type="text" style="height: auto;" data-c="(x + 3)"> <span class="review"></span></p></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>Find the values for h and k below</p><p>x² + 4x + 1 = (x - h)² + k</p></div><div class="q-answer"><p>h = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p><p>k = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>Careful x² + 4x + 1 = (x + 2)² + 1</p><p>Hence h = -2</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>Find the values of a , h and k</p><p>3x² - 12x + 4 = a(x - h)² + k</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>h = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>k = <input type="text" style="height: auto;" data-c="-8"> <span class="review"></span></p></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>Match up the function with the correct graph by dragging it in to the correct place</p><p><span class="q-text-draggable draggable" draggable="true">(x + 2)(x - 3)</span> <span class="q-text-draggable draggable" draggable="true">-(x - 2)²</span> <span class="q-text-draggable draggable" draggable="true">-(x + 3)(x - 2)</span> <span class="q-text-draggable draggable" draggable="true">(x + 2)²</span> <span class="q-text-draggable draggable" draggable="true">-(x + 2)(x - 3)</span> <span class="q-text-draggable draggable" draggable="true">-(x +2)²</span> <span class="q-text-draggable draggable" draggable="true">(x + 3)(x - 2)</span> <span class="q-text-draggable draggable" draggable="true">(x - 2)²</span></p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><p><img alt="" src="../../files/functions/quadratics/quiz/g1.jpg" style="width: 250px; height: 250px;"></p></td><td> f(x) = <input type="text" style="height: auto;" data-c="(x - 2)²"> <span class="review"></span></td><td><p><img alt="" src="../../files/functions/quadratics/quiz/g2.jpg" style="width: 250px; height: 250px;"></p></td><td> f(x) = <input type="text" style="height: auto;" data-c="-(x - 2)²"> <span class="review"></span></td></tr><tr><td><img alt="" src="../../files/functions/quadratics/quiz/g3.jpg" style="width: 250px; height: 250px;"></td><td> f(x) =<input type="text" style="height: auto;" data-c="(x + 2)²"> <span class="review"></span></td><td><img alt="" src="../../files/functions/quadratics/quiz/q6.jpg" style="width: 250px; height: 250px;"></td><td> f(x) =<input type="text" style="height: auto;" data-c="-(x + 2)(x - 3)"> <span class="review"></span></td></tr><tr><td><p><img alt="" src="../../files/functions/quadratics/quiz/g4.jpg" style="width: 250px; height: 250px;"></p></td><td> f(x) =<input type="text" style="height: auto;" data-c="(x + 3)(x - 2)"> <span class="review"></span></td><td><img alt="" src="../../files/functions/quadratics/quiz/g5.jpg" style="width: 250px; height: 250px;"></td><td> f(x) =<input type="text" style="height: auto;" data-c="(x + 2)(x - 3)"> <span class="review"></span></td></tr></tbody></table></div><div class="q-explanation"><p>f(x) = a (x - b)(x - c) has x intercepts at x = b and x = c</p><p>It is concave up when a>0 </p><p>It is concave down when a<0</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>Match up the function with the correct graph by dragging it into the correct place</p><p><span class="q-text-draggable draggable" draggable="true">(x - 2)² + 1</span> <span class="q-text-draggable draggable" draggable="true">(x - 1)² + 2</span> <span class="q-text-draggable draggable" draggable="true">(x + 2)² + 1</span> <span class="q-text-draggable draggable" draggable="true">-(x - 1)² + 2</span> <span class="q-text-draggable draggable" draggable="true">-(x - 2)² + 1</span> <span class="q-text-draggable draggable" draggable="true">(x + 1)² - 2</span> <span class="q-text-draggable draggable" draggable="true">-(x + 2)² + 1</span> <span class="q-text-draggable draggable" draggable="true">-(x - 1)² - 2</span> </p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><img alt="" src="../../files/functions/quadratics/quiz/g7.jpg" style="width: 250px; height: 250px;"></td><td>f(x) = <input type="text" style="height: auto;" data-c="(x + 2)² + 1"> <span class="review"></span> </td><td><img alt="" src="../../files/functions/quadratics/quiz/g9.jpg" style="width: 250px; height: 250px;"></td><td>f(x) = <input type="text" style="height: auto;" data-c="-(x - 2)² + 1"> <span class="review"></span></td></tr><tr><td><p><img alt="" src="../../files/functions/quadratics/quiz/g8.jpg" style="width: 250px; height: 250px;"></p></td><td>f(x) = <input type="text" style="height: auto;" data-c="-(x + 2)² + 1"> <span class="review"></span></td><td><img alt="" src="../../files/functions/quadratics/quiz/g10.jpg" style="width: 250px; height: 250px;"></td><td>f(x) = <input type="text" style="height: auto;" data-c="(x - 2)² + 1"> <span class="review"></span></td></tr><tr><td> </td><td> </td><td> </td><td> </td></tr></tbody></table></div><div class="q-explanation"><p>f(x) = a(x - h)² + k has a vertex at the point (h , k)</p><p>It is concave up when a>0 </p><p>It is concave down when a<0</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>What is the value of the discriminant in the quadratic equation 2x² - 5x + 4 = 0 ?</p></div><div class="q-answer"><p>discriminant = <input type="text" style="height: auto;" data-c="-7"> <span class="review"></span></p></div><div class="q-explanation"><p>discriminant = b² - 4ac</p><p>= (-5)² - 4x2x4</p><p>= 25 - 32</p><p>= -7</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>How many real roots does the function f have?</p><p>f(x) = 2x² + 12x + 18</p></div><div class="q-answer"><p>number of roots = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>discriminant = b² - 4ac</p><p>= (12)² - 4x2x18</p><p>= 0</p><p>Since discriminant = 0, there is one (repeated root)</p><p>We can see this if we write the function in the vertex from</p><p>f(x) = 2(x + 3)²</p><p>The vertex lies on the x axis at (-3 , 0)</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 function below is given by f(x) = ax² + bx + c</p><p>Find a , b and c</p><p><img alt="" src="../../files/functions/quadratics/quiz/g13.jpg" style="width: 350px; height: 350px;"></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="12"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p></div><div class="q-explanation"><p>Since the x intercepts (or zeros) are at x = -3 and x = -1, we know that we can write the function in the form</p><p>f(x) = a(x + 3)(x + 1)</p><p>Let's expand the brackets</p><p>f(x) = a(x² + 4x + 3)</p><p>f(x) = ax² + 4ax + 3a</p><p>Since the y intercept is at y = 9, then 3a = 9</p><p>a = 3</p><p>f(x) = 3x² + 12x + 9</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 function below is given by f(x) = ax² + bx + c</p><p>Find a , b and c</p><p><img alt="" src="../../files/functions/quadratics/quiz/g11.jpg" style="width: 350px; height: 350px;"></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="-12"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="-14"> <span class="review"></span></p></div><div class="q-explanation"><p>Using the vertex (-3 , 4), we know</p><p>f(x) = a (x + 3)² + 4</p><p>Use the other point (0 , -14) to find a</p><p>-14 = a (0 + 3)² + 4</p><p>-14 = 9a + 4</p><p>-18 = 9a</p><p>a = -2</p><p>f(x) = -2 (x + 3)² + 4</p><p>f(x) = -2(x² + 6x + 9) + 4</p><p>f(x) = -2x² - 12x - 18 + 4</p><p>f(x) = -2x² - 12x - 14</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 where f(x) = ax² + bx + c is given below</p><p>Drag the correct values into the table</p><p><img alt="" src="../../files/functions/quadratics/quiz/g12.jpg" style="width: 350px; height: 350px;"></p><p> <span class="q-text-draggable draggable" draggable="true">negative</span> <span class="q-text-draggable draggable" draggable="true">zero</span> <span class="q-text-draggable draggable" draggable="true">positive</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="positive"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="positive"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="positive"> <span class="review"></span></p><p>b² - 4ac = <input type="text" style="height: auto;" data-c="zero"> <span class="review"></span></p></div><div class="q-explanation"><p>b² - 4ac = 0 because there is one (repeated root).</p><p>c is positive. We can see the y intercept is positive.</p><p>a is positive. The graph is concave up.</p><p>b is positive. We can write the function in the vertex form f(x) = a(x - h)² . Since the vertex crosses the x axis where x<0, h must be negative. If you expand these brackets you will see that the x term is positive.</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="497"> <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>The graph below is the function f(x)=a(x-h)²+k</p> <p>The graph pass through the point (0,-9) and has vertex (2,3)</p> <p><img alt="" height="357" src="../../files/functions/quadratics/quadratics1image.jpg" width="295"></p> <div> <p>a) Write down the value of a, h and k</p> </div> <p>b) Find f(x) giving your answer in the form f(x) = Ax²+Bx+C</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">h and k give the coordinates of the vertex (h,k)<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/quadratics/esq_quadratics1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/esq_quadratics1.pdf" width="640"></iframe></p> </section> <h4> </h4> </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="496"> <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>Let f(x) = 2x²+12x +11</p> <p>The function can also be expressed in the form f(x) = a(x-h)² +k</p> <p>a) Find the equation of the axis of symmetry</p> <p>b) Write down the value of h</p> <p>c) Write down the value of k</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">h and k give the coordinates of the vertex (h,k)<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/quadratics/esq_quadratics2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/esq_quadratics2.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <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="498"> <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>Let f(x) = x² + 2px + (3p+4)</p> <p>Find the value of p so that f(x)=0 has two <strong>equal </strong>roots.</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>You can use the discriminant to solve this question</p> <p><span class="math-tex">\(\triangle =b^ 2-4ac\)</span></p> <p><content>For two <strong>equal </strong>roots , b² - 4ac = 0</content></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/quadratics/esq_quadratics3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/esq_quadratics3.pdf" width="640"></iframe></p> </section> <h4> </h4> </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="501"> <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 hard"> <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>Let <strong>f(x)=2x² + kx + 1</strong> and <strong>g(x) = -x – 1</strong>.</p> <p>The graphs of <strong>f</strong> and <strong>g</strong> intersect at two distinct points.</p> <p>Find the possible values of <strong>k</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><img alt="" height="340" src="../../files/functions/quadratics/esq-quadratics4-image.png" width="284"></p> <p>If <strong>f</strong> and <strong>g</strong> intersect at two distinct points, then <strong>f(x)=g(x) </strong>has two roots (discriminant >0)</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/quadratics/esq_quadratics4hl.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/esq_quadratics4hl.pdf" width="640"></iframe></p> </section> <h4> </h4> </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="499"> <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> <p>A quadratic function <strong><em>f</em></strong> can be written in the form <strong><em>f(x) = a(x - h)² + k </em></strong>. The graph of <strong><em>f</em></strong> has vertex <strong><em>(-1 , 4)</em></strong> and has y-intercept at <strong><em>(0 , -5)</em></strong>.</p> <p>a) Find the value of <strong><em>a</em></strong> , <strong><em>h</em></strong> and <strong><em>k</em></strong>.</p> <p>b) The line <strong><em>y = mx + 1</em></strong> is a tangent to the curve <strong><em>f</em></strong>. Find the value of <strong><em>m</em></strong>.</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><img alt="" src="../../files/functions/quadratics/esq5-image.png" style="width: 350px; height: 452px;"></p> <p>b) If the line y = mx + 1 is a tangent to the curve to the curve of <strong><em>f</em></strong>, then the solution to</p> <p>mx+1=a(x-h)²+k</p> <p>has one repeated root.</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/quadratics/esq_quadratics5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/esq_quadratics5.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-red"> <div class="panel-heading"> <div> <p>Just for Fun</p> </div> </div> <div class="panel-body"> <div> <p>If you understand the form y=a(x - p)(x - q) of a quadratic equation, then you should be able to play and win this game of angry birds. <strong>The angry bird starts at (0 , 0), the pig is at (10 , 0) and the bird must pass through (5,4).</strong><br> Enter the equation in the space below and press play.</p> <p style="text-align: center;"><iframe height="383px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/TeK9jaey/width/629/height/383/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false" style="border:0px;" title="Angry BIrds L1" width="629px"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="679"> <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>Quadratics</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"> <button id="save-my-progress" type="button" class="btn btn-sm btn-primary text-center btn-xs-block"> <i class="fa fa-fw fa-floppy-o"></i> Save </button> 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