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<title>Amplitude, Period, Phase Shift and Frequency</title>
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<h1 class="center">Amplitude, Period, Phase Shift and Frequency</h1>
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<p class="center">Some functions (like <a href="trig-sin-cos-tan-graphs.html">Sine and Cosine</a>) repeat forever<br>
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and are called <b>Periodic Functions.</b></p>
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<p>The <b>Period</b> goes from one peak to the next (or from any point to the next matching point):</p>
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<p class="center"><img src="images/period-amplitude.svg" alt="period and amplitude" height="153" width="511"></p>
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<p>The <b>Amplitude</b> is the height from the center line to the peak (or to the trough). Or we can measure the height from highest to lowest points and divide that by 2.</p>
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<p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/phase-shift.svg" alt="phase shift" height="176" width="251"></p>
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<p>The <b>Phase Shift</b> is how far the function is shifted <b>horizontally</b> from the usual position.</p>
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<div style="clear:both"></div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/vertical-shift.svg" alt="vertical shift" height="179" width="241"></p>
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<p>The <b>Vertical Shift</b> is how far the function is shifted <b>vertically</b> from the usual position.</p>
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<div style="clear:both"></div>
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<h2>All Together Now!</h2>
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<p>We can have all of them in one equation:</p>
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<p class="large center">y = A sin(B(x + C)) + D</p>
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<ul>
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<li>amplitude is <b>A</b></li>
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<li>period is <b>2<span class="times">π</span>/B</b></li>
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<li>phase shift is <b>C</b> (positive is to the <b>left</b>)</li>
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<li>vertical shift is <b>D</b></li>
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</ul>
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<p>And here is how it looks on a graph:</p>
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<p class="center"><img src="images/a-sin-bxcd.svg" alt="aa" height="215" width="355"></p>
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<p><br></p>
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<p>Note that we are using <a href="../geometry/radians.html">radians</a> here, not degrees, and there are 2<span class="times">π</span> radians in a full rotation.</p>
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<div class="example">
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<h3>Example: sin(x)</h3>
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<p>This is the basic unchanged sine formula. <b>A = 1, B = 1, C = 0 and D = 0</b></p>
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<p>So amplitude is <b>1</b>, period is <b>2<span class="times">π</span></b>, there is no phase shift or vertical shift:</p>
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<p class="center"><img src="images/amplitude-period-ex0.svg" alt="amplitude 1, period 2pi, no shifts" height="169" width="427"></p>
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</div>
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<div class="example">
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<h3>Example: 2 sin(4(x − 0.5)) + 3</h3>
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<ul>
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<li>amplitude <b>A = 2</b></li>
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<li>period <b>2<span class="times">π</span>/B</b> = <b>2<span class="times">π</span>/4 = <span class="times">π</span>/2</b></li>
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<li>phase shift<b> = −0.5</b> (or <b>0.5</b> to the right)</li>
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<li>vertical shift <b>D = 3</b></li>
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</ul>
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<p class="center"><img src="images/amplitude-period-ex2.svg" alt="amplitude 2, period pi/2, phase shift 0.5, vert shift 3" height="211" width="374"></p>
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<p>In words:</p>
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<ul>
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<li>the <b>2</b> tells us it will be 2 times taller than usual, so Amplitude = 2</li>
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<li>the usual period is 2<b><span class="times">π</span></b>, but in our case that is "sped up" (made shorter) by the <b>4</b> in 4x, so Period = <b><span class="times">π</span>/2</b></li>
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<li>and the <b>−0.5</b> means it will be shifted to the <b>right</b> by <b>0.5</b></li>
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<li>lastly the <b>+3</b> tells us the center line is y = +3, so Vertical Shift = 3</li>
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</ul>
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</div>
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<p>Instead of <b>x</b> we can have <b>t</b> (for time) or maybe other variables:</p>
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<div class="example">
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<h3>Example: 3 sin(100t + 1)</h3>
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<p>First we need brackets around the (t+1), so we can start by dividing the 1 by 100:</p>
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<p>3 sin(100t + 1) = <b>3 sin(100(t + 0.01))</b></p>
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<p>Now we can see:</p>
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<ul>
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<li>amplitude is <b>A = 3</b></li>
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<li>period is <b>2<span class="times">π</span>/100</b> = <b>0.02 <span class="times">π</span></b></li>
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<li>phase shift is <b>C</b><b> =</b> <b>0.01</b> (to the left)</li>
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<li>vertical shift is <b>D = 0</b></li>
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</ul>
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<p>And we get:</p>
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<p class="center"><img src="images/amplitude-period-ex3.svg" alt="amplitude 3, period 0.02pi, phase shift -0.01, no vertical shift" height="239" width="395"></p>
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</div>
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<h2>Frequency</h2>
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<p>Frequency is how often something happens per unit of time (per "1").</p>
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<div class="example">
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<h3>Example: Here the cosine function repeats 4 times between 0 and 1:</h3>
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<p class="center"><img src="images/period-frequency.svg" alt="period 1/4, frequency 4" height="244" width="425"></p>
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<p>So the Frequency is 4</p>
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<p>And the Period is
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<span class="intbl">
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<em>1</em>
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<strong>4</strong>
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</span></p>
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</div>
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<p>In fact the Period and Frequency are related:</p>
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<p class="large center">Frequency = <span class="intbl">
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<em>1</em>
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<strong>Period</strong>
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</span></p>
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<p class="large center">Period = <span class="intbl">
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<em>1</em>
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<strong>Frequency</strong>
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</span></p>
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<div class="example">
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<h3>Example from before: 3 sin(100(t + 0.01))</h3>
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<p class="center"><img src="images/amplitude-period-ex3.svg" alt="amplitude 3, period 0.02pi, phase shift -0.01, no vertical shift" height="239" width="395"></p>
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<p class="center">The period is 0.02<b><span class="times">π</span></b></p>
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<p class="center">So the Frequency is
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<span class="intbl">
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<em>1</em>
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<strong>0.02<span class="times">π</span></strong>
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</span> =
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<span class="intbl">
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<em>50</em>
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<strong><span class="times">π</span></strong>
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</span></p>
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</div>
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<p>Some more examples:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th align="center">Period</th>
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<th align="center">Frequency</th>
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</tr>
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<tr>
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<td style="text-align:center;"><span class="intbl">
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<em>1</em>
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<strong>10</strong>
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</span></td>
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<td style="text-align:center;">10</td>
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</tr>
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<tr>
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<td style="text-align:center;"><span class="intbl">
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<em>1</em>
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<strong>4</strong>
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</span></td>
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<td style="text-align:center;">4</td>
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</tr>
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<tr>
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<td style="text-align:center;">1</td>
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<td style="text-align:center;">1</td>
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</tr>
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<tr>
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<td style="text-align:center;">5</td>
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<td style="text-align:center;"><span class="intbl">
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<em>1</em>
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<strong>5</strong>
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</span></td>
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</tr>
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<tr>
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<td style="text-align:center;">100</td>
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<td style="text-align:center;"><span class="intbl">
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<em>1</em>
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<strong>100</strong>
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</span></td>
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</tr>
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</tbody></table>
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</div>
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<p>When frequency is <b>per second</b> it is called "Hertz".</p>
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<div class="example">
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<h3>Example: 50 Hertz means 50 times per second</h3>
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</div>
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<p class="center"><img src="images/motocross-bouncing.jpg" alt="motocross bouncing" height="234" width="249"><br>
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The faster it bounces the more it "Hertz"!</p>
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<h2>Animation</h2>
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<div class="script" style="height: 390px;">
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../algebra/images/wave-sine.js
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</div>
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<p> </p>
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<div class="questions">7784,7785,7788,7789,9863,7793,7794,7795,7796,7792</div>
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<div class="related">
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<a href="trigonometry.html">Trigonometry</a>
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<a href="index.html">Algebra Index</a>
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</div>
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