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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="../540/geometry-trigonometry.html">Geometry & Trigonometry</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Trigonometric Graphs</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/trigonometry/trig-graphs/main-1.png" style="float: left; width: 100px; height: 100px;"> In this page, we will learn about the graphs of the circular functions sinx, cosx and tanx. We will link this to work that we have done on <a href="../677/transforming-functions-sl.html" title="Transforming Functions SL">Transforming Functions</a> to see how translating and stretching the graphs affect the tigonometric functions. It is fairly common to get a long question on this topic in the exam and they can come up on paper 1 or paper 2, so you need to be comfortable using your calculator and working without it. You can find typical questions in the exam-style questions section.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn about</p> <ul> <li>the graphs of the functions sinx, cosx and tanx</li> <li>the amplitude and period of a trigonometric graph</li> <li>Composite functions of the form <span class="math-tex">\(\large f(x) = a \sin(b(x+c))+d\)</span> and <span class="math-tex">\(\large f(x) = a \cos(b(x+c))+d\)</span></li> <li>Transformations of trigonometric functions</li> <li>Modelling with trigonometric functions</li> </ul> </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 frameborder="0" height="480" scrolling="no" src="../../files/trigonometry/trig-graphs/revision-notes_trig-graphs.pdf" width="100%"></iframe></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 about the period of a trigonometric function</p> <hr class="hidden-separator"> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#45c99af3"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="45c99af3"> <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%;"> Period of Trig Function <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="11-922-2825" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>What is the period of this function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-period/q1.png" style="width: 450px; height: 356px;"></p></div><div class="q-answer"><p>period = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-period/q1_ans.png" style="width: 400px; height: 317px;"></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> f(x) = sin 2x</p><p>What is the period of the function <strong><em>f</em></strong> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2\pi\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 4\pi\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \pi\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{\pi}{2}\)</span></label></p></div><div class="q-explanation"><p>The function f(x) = sin(<em><strong>a</strong></em>x) has period <span class="math-tex">\(\large \frac{2\pi}{a}\)</span></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> f(x) = cos 0.5x</p><p>What is the period of the function <strong><em>f</em></strong> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2\pi\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \pi\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large 4\pi\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{\pi}{2}\)</span></label></p></div><div class="q-explanation"><p>The function f(x) = cos(<em><strong>a</strong></em>x) has period <span class="math-tex">\(\large \frac{2\pi}{a}\)</span></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> <span class="math-tex">\(\large f(x) = \sin \pi x\)</span></p><p>What is the period of the function <strong><em>f</em></strong> ?</p></div><div class="q-answer"><p>period = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>The function f(x) = sin(<em><strong>a</strong></em>x) has period <span class="math-tex">\(\large \frac{2\pi}{a}\)</span></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> <span class="math-tex">\(\large f(x) = \cos \frac{\pi x}{2}\)</span></p><p>What is the period of the function <strong><em>f</em></strong> ?</p></div><div class="q-answer"><p>period = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>The function f(x) = cos(<em><strong>a</strong></em>x) has period <span class="math-tex">\(\large \frac{2\pi}{a}\)</span></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> <span class="math-tex">\(\large f(x) = \sin 4\pi x\)</span></p><p>What is the period of the function <strong><em>f</em></strong> ?</p></div><div class="q-answer"><p>period = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p></div><div class="q-explanation"><p>The function f(x) = sin(<em><strong>a</strong></em>x) has period <span class="math-tex">\(\large \frac{2\pi}{a}\)</span></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> <span class="math-tex">\(\large f(x) = 2\cos \frac{\pi x}{15}\)</span></p><p>What is the period of the function <strong><em>f</em></strong> ?</p></div><div class="q-answer"><p>period = <input type="text" style="height: auto;" data-c="30"> <span class="review"></span></p></div><div class="q-explanation"><p>The function f(x) = cos(<em><strong>a</strong></em>x) has period <span class="math-tex">\(\large \frac{2\pi}{a}\)</span></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> <span class="math-tex">\(\large f(x) = \sin (\frac{2\pi }{5}(x-5))\)</span></p><p>What is the period of the function <strong><em>f</em></strong> ?</p></div><div class="q-answer"><p>period = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>The function <span class="math-tex">\(\large f(x) = \sin (a(x-b))\)</span> has period <span class="math-tex">\(\large \frac{2\pi}{a}\)</span></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 equation for this function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-period/q9.png" style="width: 400px; height: 317px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=\cos(\frac{\pi}{20}x)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=\cos(\frac{\pi}{5}x)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=\cos(10{\pi}x)\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large f(x)=\cos(\frac{\pi}{10}x)\)</span></label></p></div><div class="q-explanation"><p>The function f(x) = cos(<em><strong>a</strong></em>x) has period <span class="math-tex">\(\large \frac{2\pi}{a}\)</span></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 equation for this function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-period/q10_1.png" style="width: 400px; height: 317px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=\sin(\frac{\pi}{15}x)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=\sin(\frac{15}{2\pi}x)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=\sin(15x)\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large f(x)=\sin(\frac{2\pi}{15}x)\)</span></label></p></div><div class="q-explanation"><p>The function f(x) = sin(<em><strong>a</strong></em>x) has period <span class="math-tex">\(\large \frac{2\pi}{a}\)</span></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> <hr class="hidden-separator"> <p>Here is a quiz about the amplitude of a trigonometric function</p> <hr class="hidden-separator"> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#40eeaf95"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="40eeaf95"> <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%;"> Amplitude of a Trigonometric Function <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="11-923-2825" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>What is the amplitude of this function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q1.png" style="width: 400px; height: 299px;"></p></div><div class="q-answer"><p>amplitude = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q1a.png" style="width: 400px; height: 299px;"></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 amplitude of this function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q2.png" style="width: 400px; height: 299px;"></p></div><div class="q-answer"><p>amplitude = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q2ans.png" style="width: 400px; height: 299px;"></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 amplitude of this function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q3.png" style="width: 400px; height: 299px;"></p></div><div class="q-answer"><p>amplitude = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q3ans.png" style="width: 400px; height: 299px;"></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 amplitude of this function?</p><p><span class="math-tex">\(\large f(x)=2\cos 3x\)</span></p></div><div class="q-answer"><p>amplitude = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>The function, f(x) = a cosbx has amplitude |a|</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 amplitude of this function?</p><p><span class="math-tex">\(\large f(x)=-3\sin 0.5x\)</span></p></div><div class="q-answer"><p>amplitude = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>The function, f(x) = a sin bx has amplitude |a|</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 diagram shows the graph of <strong><em>f</em></strong>.</p><p>What is the value of <strong><em>a</em></strong> in the following function? <span class="math-tex">\(\large f(x)= a\cos(x-\frac{\pi}{3})+1\)</span></p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q6.png" style="width: 400px; height: 269px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q6ans_1.png" style="width: 400px; height: 269px;"></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 diagram shows the graph of <strong><em>f</em></strong>.</p><p>What is the value of <strong><em>a</em></strong> in the following function? <span class="math-tex">\(\large f(x)= a\sin(5x)+4\)</span></p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q7.png" style="width: 400px; height: 269px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q7ans.png" style="width: 400px; height: 269px;"></p><p>The amplitude = 3</p><p>However, the graph of <em><strong>y = 3 sin(5x)</strong></em> has been reflected in the x axis, so <em><strong>f(x) = -3 sin(5x)</strong></em></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p> Which is the correct graph of the function <span class="math-tex">\(\large f(x) = -1.5\sin(x+\frac{\pi}{2})+1.5\)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: center;">A</td><td style="text-align: center;">B</td><td style="text-align: center;">C</td><td style="text-align: center;">D</td></tr><tr><td style="text-align: center;"><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q8a.png" style="width: 150px; height: 101px;"></td><td style="text-align: center;"><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q8b.png" style="width: 150px; height: 101px;"></td><td style="text-align: center;"><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q8c.png" style="width: 150px; height: 101px;"></td><td style="text-align: center;"><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q8d.png" style="width: 150px; height: 101px;"></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> D</label></p></div><div class="q-explanation"><p><span class="math-tex">\(\large f(x) = -1.5\sin(x+\frac{\pi}{2})+1.5\)</span></p><p>Start with graph of y = sinx</p><ol><li>Translate left <span class="math-tex">\(\large\frac{\pi}{2}\)</span></li><li>Stretch by a factor 1.5 in the y direction</li><li>Reflect in x axis</li><li>Translate up 1.5 units</li></ol></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 <strong><em>f</em></strong><em> </em>is plotted below. Which is the correct function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q9.png" style="width: 400px; height: 271px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x) = -7\cos(2(x-\frac{\pi}{2}))+1.5\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x) = 3.5\cos(2(x-\frac{\pi}{2}))+1.5\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large f(x) = -3.5\cos(2(x-\frac{\pi}{2}))+1.5\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x) = 7\cos(2(x-\frac{\pi}{2}))+1.5\)</span></label></p></div><div class="q-explanation"><p> The amplitude is 3.5 and the graph of cos x has been reflected in the x axis. Hence a = -3.5</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 <strong><em>f</em></strong><em> </em>is plotted below. Which is the correct function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-amplitude/q10_1.png" style="width: 400px; height: 235px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x) = 2\sin(\pi(x+0.5))-1\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x) =-2.5\sin(\pi(x+0.5))+3\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large f(x) = 2.5\sin(\pi(x+0.5))-3\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x) = 3\sin(\pi(x+0.5))-2\)</span></label></p></div><div class="q-explanation"><p>The amplitude is 2.5</p><p><em><strong>a </strong></em>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> <hr class="hidden-separator"> <p>Here is a quiz about translations of a trigonometric function</p> <hr class="hidden-separator"> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#e53006fd"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="e53006fd"> <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%;"> Translating Trigonometric Functions <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-924-2825" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p> The graph of <strong><em>f</em></strong> is shown below. Which is the correct function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q1.png" style="width: 400px; height: 205px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=2 \sin(x+\pi)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=2 \sin(x+\frac{\pi}{2})\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=2 \sin(x-\pi)\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large f(x)=2 \sin(x-\frac{\pi}{2})\)</span></label></p></div><div class="q-explanation"><p>The graph of y = 2 sinx is translated right <span class="math-tex">\(\large \frac{\pi}{2}\)</span></p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q1ans.png" style="width: 400px; height: 205px;"></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 <strong><em>f</em></strong> is shown below. Which is the correct function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q2.png" style="width: 400px; height: 182px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=-0.5 \cos(x-\frac{\pi}{6})\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=-0.5 \cos(x+\frac{\pi}{6})\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=-0.5 \cos(x-\frac{\pi}{3})\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large f(x)=-0.5 \cos(x+\frac{\pi}{3})\)</span></label></p></div><div class="q-explanation"><p>The graph of y = -0.5 cosx is translated left <span class="math-tex">\(\large \frac{\pi}{3}\)</span></p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q2ans.png" style="width: 400px; height: 182px;"></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 diagram below shows the graph of <span class="math-tex">\(\large f(x) = 2\sin x+a\)</span></p><p>What is the value of <strong><em>a</em></strong>?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q3.png" style="width: 400px; height: 206px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>The graph of y = 2 sinx has been translated up 1 unit</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q3ans.png" style="width: 400px; height: 206px;"></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 diagram below shows the graph of <span class="math-tex">\(\large f(x) = -3\cos x +a\)</span></p><p>What is the value of <strong><em>a</em></strong>?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q4.png" style="width: 400px; height: 207px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>The graph of y = -3cosx is translated up 3 units</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q4ans.png" style="width: 400px; height: 207px;"></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 diagram below shows the graph of <span class="math-tex">\(\large f(x) = 0.5\sin 2x +a\)</span></p><p>What is the value of <strong><em>a</em></strong>?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q5.png" style="width: 400px; height: 207px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-1.5"> <span class="review"></span></p></div><div class="q-explanation"><p>The graph of y =0.5sin2x is translated down 1.5 units</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q5ans.png" style="width: 400px; height: 207px;"></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 diagram below shows the graph of <span class="math-tex">\(\large f(x) =-2.5\sin(\frac{\pi}{5}x)+a\)</span></p><p>What is the value of <strong><em>a</em></strong>?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q6_1.png" style="width: 400px; height: 284px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="3.5"> <span class="review"></span></p></div><div class="q-explanation"><p>The graph of <span class="math-tex">\(\large y =-2.5\sin(\frac{\pi}{5}x)+a\)</span> is translated up 3.5 units</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q6_1ans.png" style="width: 400px; height: 284px;"></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 <strong><em>f</em></strong> is shown below. Which is the correct function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q7.png" style="width: 400px; height: 237px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)= \sin(\pi(x-1))\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)= \sin(\pi(x+1))\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)= \sin(\pi(x-0.5))\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large f(x)= \sin(\pi(x+0.5))\)</span></label></p></div><div class="q-explanation"><p>The graph of <span class="math-tex">\(\large y = \sin\pi x\)</span> is translated left by 1 unit</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q7ans.png" style="width: 400px; height: 251px;"></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 <strong><em>f</em></strong> is shown below. Which is the correct function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q8.png" style="width: 400px; height: 289px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x) = \cos\frac{\pi}{2}( x+1)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x) = \cos\frac{\pi}{2}( x-1)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x) = \cos\frac{\pi}{2}( x+0.5)\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large f(x) = \cos\frac{\pi}{2}( x-0.5)\)</span></label></p></div><div class="q-explanation"><p>The graph of <span class="math-tex">\(\large y = \cos\frac{\pi}{2} x\)</span> is translated right <span class="math-tex">\(\large \frac{1}{2}\)</span></p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q8_1ans.png" style="width: 400px; height: 274px;"></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 <strong><em>f</em></strong> is shown below. Which is the correct function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q9.png" style="width: 400px; height: 274px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=-2 \cos(\frac{\pi}{5}(x+3))\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=-2 \cos(\frac{\pi}{5}(x-3))\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=-2 \cos(\frac{\pi}{5}(x+2))\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large f(x)=-2 \cos(\frac{\pi}{5}(x-2))\)</span></label></p></div><div class="q-explanation"><p>The graph of <span class="math-tex">\(\large y=-2 \cos(\frac{\pi}{5}x)\)</span> is translated right 2</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q9ans.png" style="width: 400px; height: 287px;"></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 <strong><em>f</em></strong> is shown below. Which is the correct function?</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q10.png" style="width: 400px; height: 223px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=3 \sin(\frac{\pi}{3}(x-3))\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=3 \sin(\frac{\pi}{3}(x+4.5))\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large f(x)=3 \sin(\frac{\pi}{3}(x-1.5))\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large f(x)=3 \sin(\frac{\pi}{3}(x+1.5))\)</span></label></p></div><div class="q-explanation"><p>The graph of <span class="math-tex">\(\large y=3 \sin(\frac{\pi}{3}x)\)</span> is translated left 1.5</p><p><img alt="" src="../../files/trigonometry/trig-graphs/quiz-translations/q10ans.png" style="width: 400px; height: 223px;"></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> <hr class="hidden-separator"></div> </div> <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>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div> <div class="panel 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="943"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The height <strong><em>h</em></strong> of water, in metres, in a habour is modelled by the function <span class="math-tex">\(\large h(t)=5.5\sin(0.5(t-1.5))+12\)</span> where <strong><em>t</em></strong> is time after midday in hours.</p> <p>a) Find the initial height of the water.</p> <p>b) At what time is it when the water reaches this height again?</p> <p>c) Find the maximum height of the water.</p> <p>d) How much time is there in between the first and second time that the water at 16 metres?</p> <p>Give heights to 3 significant figures and times to the nearest minute</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>This question gets you to use your calculator to solve problems.</p> <p>Ensure that your calculator is in radian mode.</p> <p>Adjust the view window so that you get a good view of the graph</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq2h1.png" style="width: 268px; height: 154px;"></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>a) 8.25 metres</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq2_1.png" style="width: 268px; height: 154px;"></p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq2_2.png" style="width: 268px; height: 154px;"> <img alt="" src="../../files/trigonometry/trig-graphs/esq2_3.png" style="width: 268px; height: 154px;"></p> <p>b) 21:17 or 9.17 pm</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq2_4.png" style="width: 268px; height: 154px;"> 0.283x60 = 17 minutes</p> <p>c) 4.64 meters</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq2_5.png" style="width: 268px; height: 154px;"></p> <p>d) 3 hours 2 mins</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq2_6.png" style="width: 268px; height: 154px;"> <img alt="" src="../../files/trigonometry/trig-graphs/esq2_7.png" style="width: 268px; height: 154px;"></p> <p>6.155 - 3.129 = 3.026</p> <p>0.026 x 60 = 2 mins</p> </section> <h4> </h4> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-default panel-expandable panel-has-border"> <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="942"> <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-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The following diagram shows a Ferris wheel.</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/ferris2.png" style="width: 250px; height: 257px;"></p> <p>The height, <strong><em>h</em></strong> metres of a seat above ground after <strong><em>t</em></strong> minutes is given by <span class="math-tex">\(\large h(t)=a\ \cos(bt)+c\)</span> , where <strong><em>a</em></strong>, <strong><em>b</em></strong> and <strong><em>c <span class="math-tex">\(\large \in \mathbf{R}\)</span></em></strong></p> <p>The following graph shows the height of the seat.</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq1.png" style="width: 500px; height: 383px;"></p> <p>Find the values of <strong><em>a</em></strong>, <strong><em>b</em></strong> and <strong><em>c</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>amplitude is <strong><em>|a|</em></strong></p> <p><content></content>period = <span class="math-tex">\(\large \frac{2\pi}{b}\)</span></p> <p>vertical shift = <strong><em>c</em></strong></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><img alt="" src="../../files/trigonometry/trig-graphs/esq1ans.png" style="width: 400px; height: 317px;"></p> <p>amplitude is <strong><em>|a|</em></strong><em> = 6 , <strong>a = -6</strong></em></p> <p><content></content>period = <span class="math-tex">\(\large \frac{2\pi}{b}=5\\ \large b = \frac{2\pi}{5}\)</span></p> <p>vertical shift = <strong><em>c = 8</em></strong></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-expandable panel-has-border"> <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="944"> <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-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Consider a function <strong><em>f</em></strong>, such that <span class="math-tex">\(\large f(x) = 5.6\cos(\frac{\pi}{a}(x-1))+b\)</span> , <span class="math-tex">\(\large 0\le x\le 15\)</span>, <span class="math-tex">\(\large a,b\in \mathbf{R}\)</span></p> <p>The function <strong><em>f</em></strong> has a local maximum at the point (1 , 8.8) and a local minimum at the point (10 , -2.4)</p> <p>a) Find the period of the function</p> <p>b) <strong>Hence</strong>, find the value of <strong><em>a</em></strong>.</p> <p>c) Find the value of <strong><em>b</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>For <span class="math-tex">\(\large f(x)=a \cos(b(x+c))+d\)</span></p> <p>period = <span class="math-tex">\(\large \frac{2\pi}{b}\)</span><content></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>a) period = 18</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq3.png" style="width: 500px; height: 223px;"></p> <p>b) period = <span class="math-tex">\(\large \frac{2\pi}{\frac{\pi}{a}}\)</span></p> <p><span class="math-tex">\(\large \frac{2\pi}{\frac{\pi}{a}}=18\\ \large 2a = 18\\ \large a = 9\)</span></p> <p>c) Use (1 , 8.8)</p> <p><span class="math-tex">\(\large5.6\cos(\frac{\pi}{9}(1-1))+b=8.8\\ \large5.6\cos(0)+b=8.8\\ \large5.6+b=8.8\\ \large b=3.2\)</span></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-expandable panel-has-border"> <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="945"> <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>Consider a function <strong><em>f</em></strong>, such that <span class="math-tex">\(\large f(x) = a\sin(\frac{\pi}{15}(x+2))+b\)</span> , <span class="math-tex">\(\large a,b\in \mathbf{R}\)</span></p> <p>The function <strong><em>f</em></strong> has passes through the points (10.5 , 5.5) and (15.5 , 2.5)</p> <p>Find the value of <em><strong>a</strong></em> and the value of <strong><em>b</em></strong></p> <p> </p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <p> </p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>Substitute the values (10.5 , 5.5) and (15.5 , 2.5) into the function.</p> <p>This will give you two equations for the unknowns <strong><em>a</em></strong> and <strong><em>b</em></strong>.</p> <p>Solve these equations</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="math-tex">\(\large f(x) = a\cos(\frac{\pi}{15}(x+2))+b\)</span></p> <hr class="hidden-separator"> <p>(10.5 , 5.5)</p> <p><span class="math-tex">\(\large a\sin(\frac{\pi}{15}(10.5+2))+b=5.5\\ \large a\sin(\frac{12.5}{15}\pi)+b=5.5\\ \large a\sin(\frac{25}{30}\pi)+b=5.5\\ \large a\sin(\frac{5}{6}\pi)+b=5.5\\ \large a(0.5)+b=5.5\\ \large a+2b=11\)</span></p> <hr class="hidden-separator"> <p>(15.5 , 2.5)</p> <p><span class="math-tex">\(\large a\sin(\frac{\pi}{15}(15.5+2))+b=2.5\\ \large a\sin(\frac{17.5}{15}\pi)+b=2.5\\ \large a\sin(\frac{35}{30}\pi)+b=2.5\\ \large a\sin(\frac{7}{6}\pi)+b=2.5\\ \large a(-0.5)+b=2.5\\ \large -a+2b=5\)</span></p> <hr class="hidden-separator"> <p>a + 2b = 11</p> <p>-a + 2b = 5</p> <hr class="hidden-separator"> <p>4b = 16</p> <p><strong>b = 4</strong></p> <p><strong>a= 3</strong></p> <hr class="hidden-separator"></section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-expandable panel-has-border"> <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="946"> <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>The following diagram shows a ball attached to the end of a spring.<br> <img alt="" src="../../files/trigonometry/trig-graphs/esq5.png" style="width: 350px; height: 392px;"></p> <p>The height, <strong><em>h</em></strong>, in mtres of the ball above the ground <strong><em>t</em></strong> seconds after being released can be modelled by the function</p> <p><span class="math-tex">\(\large h(t)=a\cos(\frac{\pi}{b}t)+c\)</span> , <span class="math-tex">\(\large a,b, c\in \mathbf{R}\)</span></p> <p>The ball is release from an initial height of 4 metres.</p> <p>After <span class="math-tex">\(\large \frac{4}{3}\)</span> seconds, the ball is at a height of 1.6 metres.</p> <p>It takes the ball 4 seconds to return to its initial height.</p> <p>Find the values of <strong><em>a</em></strong>, <strong><em>b</em></strong> and <strong><em>c</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>If it takes 4 seconds to return to the initial height, then the period = 4</p> <p><content>Use this information to work out <em><strong>b</strong></em></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>If it takes 4 seconds to return to the initial height, then the period = 4</p> <p><span class="math-tex">\(\large \frac{2\pi}{\frac{\pi}{b}}=4\\ \large b=2\)</span></p> <hr class="hidden-separator"> <p><span class="math-tex">\(\large h(t)=a\cos(\frac{\pi}{2}t)+c\)</span></p> <p>When t = 0, h = 4</p> <p><span class="math-tex">\(\large a\cos(0)+c=4\\ \large a+c=4\)</span></p> <hr class="hidden-separator"> <p>When t = <span class="math-tex">\(\large \frac{4}{3}\)</span>, h = 1.6</p> <p><span class="math-tex">\(\large a\cos(\frac{\pi}{2}(\frac{4}{3}))+c=1.6\\ \large a\cos(\frac{2\pi}{3})+c=1.6\\ \large a(-0.5)+c=1.6\\ \large -a+2c=3.2\)</span></p> <hr class="hidden-separator"> <p>a + c = 4</p> <p>-a + 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