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onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol 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">2.5 Quadratic Models</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"> <h2><img alt="" src="../../files/MM/QM/QM-1.jpeg" style="width: 100px; height: 100px; float: left;">Parabolas</h2> <p>Quadratic functions are in the form <em>ax<sup>2</sup> + bx + c</em>, and have fascinated mathematicians for centuries. They have some fundamental properties that will explored in this unit. Probably the best known example of quadratic models are for projectiles. Anything that is thrown/launched and is subject to the laws of gravity will likely follow a quadratic path.</p> <hr> <div class="panel panel-has-colored-body panel-turquoise"> <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><span id="docs-internal-guid-9601b9f3-7fff-c187-0013-45157e5b706c"></span></p> <p dir="ltr"><span id="docs-internal-guid-9601b9f3-7fff-c187-0013-45157e5b706c">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"><span id="docs-internal-guid-9601b9f3-7fff-c187-0013-45157e5b706c"></span>Understand the general expression of a quadratic function <span class="math-tex">\(y=ax^{2}+bx+c\)</span> and understand the significance of the coefficients.</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Understand that quadratic functions model situations where there is a constant rate of change</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Derive and use quadratic models<span id="docs-internal-guid-9601b9f3-7fff-c187-0013-45157e5b706c"></span></p> </li> </ul> </div> </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> <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>Slides Gallery</p> </div> </div> <div class="panel-body"> <div> <p>Use these slides to review the material and key points covered in the videos.</p> <div id="carousel-244" class="dynamic-gallery carousel slide" data-id="244"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-244/quadratics-no-logos001.jpeg" data-fancybox="gallery-244" title="" data-caption=""><img alt="" src="../../../std-galleries/15-244/quadratics-no-logos001.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-244/quadratics-no-logos002.jpeg" data-fancybox="gallery-244" title="" data-caption=""><img alt="" src="../../../std-galleries/15-244/quadratics-no-logos002.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-244/quadratics-no-logos003.jpeg" data-fancybox="gallery-244" title="" data-caption=""><img alt="" src="../../../std-galleries/15-244/quadratics-no-logos003.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-244/quadratics-no-logos004.jpeg" data-fancybox="gallery-244" title="" data-caption=""><img alt="" src="../../../std-galleries/15-244/quadratics-no-logos004.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-244/quadratics-no-logos005.jpeg" data-fancybox="gallery-244" title="" data-caption=""><img alt="" src="../../../std-galleries/15-244/quadratics-no-logos005.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-244/quadratics-no-logos006.jpeg" data-fancybox="gallery-244" title="" data-caption=""><img alt="" src="../../../std-galleries/15-244/quadratics-no-logos006.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-244/quadratics-no-logos007.jpeg" data-fancybox="gallery-244" title="" data-caption=""><img alt="" src="../../../std-galleries/15-244/quadratics-no-logos007.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-244" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-244" 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/15-244/quadratics-no-logos001-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-244/quadratics-no-logos002-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-244/quadratics-no-logos003-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/15-244/quadratics-no-logos004-thumb128.jpg"><li><li data-index="4"><img title="Click to view" src="../../../std-galleries/15-244/quadratics-no-logos005-thumb128.jpg"><li><li data-index="5"><img title="Click to view" src="../../../std-galleries/15-244/quadratics-no-logos006-thumb128.jpg"><li><li data-index="6"><img title="Click to view" src="../../../std-galleries/15-244/quadratics-no-logos007-thumb128.jpg"><li></li></ol> </div> </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>1. About quadratic</p> </div> </div> <div class="panel-body"> <div> <p>This is a general introduction to quadratic functions</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/144754956"></iframe></div> </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 points</p> </div> </div> <div class="panel-body"> <div> <p>Here we cover the key points and features of quadratic models.</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/144756241"></iframe></div> </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.The y - intercept</p> </div> </div> <div class="panel-body"> <div> <p>A short look at understanding y - intercepts and how to find them.</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/592195846"></iframe></div> </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.Axis of symmetry and vertex</p> </div> </div> <div class="panel-body"> <div> <p>This covers the ways of finding the axis of symmetry and the coordinates of the vertex of a quadratic model.<iframe allowfullscreen="" frameborder="0" height="318" mozallowfullscreen="" src="https://player.vimeo.com/video/145127217" 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. Working with functions on your GDC</p> </div> </div> <div class="panel-body"> <div> <p>This video uses the context of quadratic functions to cover the key GDC skills needed for working with functions.</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/592197265"></iframe></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" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p>This section of the page can be used for quick review. The flashcards help you go over key points and the quiz lets you practice answering questions on this subtopic.</p> <p>Flash Card</p> <p>Review this condensed 'key point' flashcard to help you check and keep ideas fresh in your mind.</p> <p style="text-align: center;"><img alt="" src="../../files/MM/QM/FC.jpg" style="width: 700px; height: 500px;"></p> </div> </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 hints where you got stuck. If you find there are still some gaps in your understanding then go back to the videos and slides above.</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="15-317-2256"><p>For the first seven questions of the is queuiz you need to refer to the quadratic function y = x<sup>2</sup> -2x - 15 and its graph, drawn below,</p><p style="text-align: center;"><img alt="" src="../../files/MM/QM/Quiz1.jpg" style="width: 500px; height: 375px;"></p><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the exact value of y, when x = 2.5? (do not include spaces in your answer)</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-13.75"> <span class="review"></span> </p></div><div class="q-explanation"><p>Type the function in to your GDC, use the table menu, set the table to go up in steps of 0.5 and read the answer from the table. OR, use the the 'value' or 'y solve' tool on your GDC.</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><p>......</p><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the exact value of x, when y = 128? (do not include spaces in your answer)</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="13"> <span class="review"></span> </p></div><div class="q-explanation"><p>Type the function in to your GDC, use the table menu, set the table appropriately and read the answer from the table. OR, use the 'x solve' tool on your GDC.</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 the following is the correct factorisation of the given quadratic?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> y = (x + 3)(x - 5)</label></p><p><label class="checkbox"><input type="checkbox"> y = (x - 3)(x +5)</label></p><p><label class="checkbox"><input type="checkbox"> y = (x + 3)(x + 5)</label></p><p><label class="checkbox"><input type="checkbox"> y = (x - 3)(x - 5)</label></p></div><div class="q-explanation"><p>Expand the brackets to be sure. SInce the coefficient of x is -2, this must come from +3, -5. SInce the value of 'c' is - 15, this corresponds.</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>Use your GDC to find the 'zeros' or 'roots' of this quadratic equation. What are the coordinates of the positive root? Answer in the form (x,y) with no spaces.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="(5,0)"> <span class="review"></span> </p></div><div class="q-explanation"><p>USe the 'Calc' or 'gsolve' menu on your calculator and choose 'zeros' or 'roots'.</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>What are the coordinates of the y-intercept of the function? Answer in the form (x,y) with no spaces.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="(0,-15)"> <span class="review"></span> </p></div><div class="q-explanation"><p>The y - intercept occurs when x = 0. This leaves y = -15.</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>What is the equation of the axis of symmetry for this function? (no spaces)</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="x=1"> <span class="review"></span> </p></div><div class="q-explanation"><p>The axis of symmetry is given by x = -b/2a, where y = ax<sup>2</sup> + bx + c</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>Use your GDC, or otherwise, to work out the coordinates of the vertex of the function. (no spaces)</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="(1,-16)"> <span class="review"></span> </p></div><div class="q-explanation"><p>Ues the 'calc' or 'gsolve' menu on your calculator and choose 'min'. Alternatively, sonce you know the axis of symmetry is x = 1, work out the value of y when x = 1.</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>Which of the following functions is shown in the diagram below?</p><p style="text-align: center;"><img alt="" src="../../files/MM/QM/Quiz2.jpg" style="width: 400px; height: 408px;"></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> 2x<sup>2</sup> + 3x - 5</label></p><p><label class="checkbox"><input type="checkbox"> x<sup>2</sup> + 3x - 5</label></p><p><label class="checkbox"><input type="checkbox"> 2x<sup>2</sup> - 3x + 5</label></p><p><label class="checkbox"><input type="checkbox"> x<sup>2</sup> - 3x + 5</label></p></div><div class="q-explanation"><p>We can see that the function intersects th y axis at (0,-5) and so c = -5. This narrows it down to 2 functions. You can see the the function has zeros at x = -2.5 and 1. These values of x should give y = 0 when substituted. </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>The profit (P) in dollars made by a given enterprise is given by <img align="middle" alt="negative 1 over 10 x squared plus 4 x minus 15" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»10«/mn»«/mfrac»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«mo»-«/mo»«mn»15«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage.php?formula=c8ca3b60e9becb56817f98ae6f457139.png"> where x is the number units sold. What was the cost of their intial investment? How many units need to be sold before they break even. (this means they have covered their costs - answer should be a whole number)</p></div><div class="q-answer"><p>Cost of initial investment <input type="text" style="height: auto;" data-c="15"> <span class="review"></span> dollars</p><p>Number of units sold to break even <input type="text" style="height: auto;" data-c="5"> <span class="review"></span> units</p></div><div class="q-explanation"><p>The initial investment is the money they spent before anything was sold. This is the y value when x = 0 , in other words, the y-intercept. The number of units for breakeven is the value of the first zero. Up until that point, the profit is negative. 5 is the fisrt whole number x - value that yields a profit of greater than or equal to zero.</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>The profit (P) in dollars made by a given enterprise is given by <img align="middle" alt="negative 1 over 10 x squared plus 4 x minus 15" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»10«/mn»«/mfrac»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«mo»-«/mo»«mn»15«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage.php?formula=c8ca3b60e9becb56817f98ae6f457139.png"> where x is the number units sold. What is the maximum profit that can be made with this model? How many units need to be sold for this maximum profit?</p></div><div class="q-answer"><p>Maximum profit <input type="text" style="height: auto;" data-c="25"> <span class="review"></span> </p><p>Number of units <input type="text" style="height: auto;" data-c="20"> <span class="review"></span> </p></div><div class="q-explanation"><p>The maximum profit is the y coordinate of the maximum point on the curve. The number of units is the corresponding x coordinate. Use your GDC to find these values.</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 class="panel-footer"> <div> </div> </div> </div> <h3> </h3> <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-has-colored-body panel-default"> <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="../../files/MM/QM/QMQ1.jpg" style="width: 700px; height: 178px;"></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/669174769"></iframe></div> <p> </p> </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><strong>Question 2</strong></p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../files/MM/QM/QMQ2.jpg" style="width: 700px; height: 252px;"></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/592200752"></iframe></div> </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"> <p><img alt="" 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