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198 lines
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<article id="content" role="main">
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<h1 class="center">Symmetry in Equations</h1>
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<p>Equations can have <a href="../geometry/symmetry.html">symmetry</a>:</p>
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<div class="center">
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<div class="boxa">
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<p><img src="images/equn-symmetry-x2.gif" alt="x^2" height="134" width="237" ><span class="larger"><br>
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Graph of x<sup>2</sup></span><br>
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Symmetry about y-axis</p>
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</div>
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<div class="boxa">
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<p><img src="images/equn-symmetry-1-x.gif" alt="x^2" height="160" width="160" ><span class="larger"><br>
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Graph of 1/x</span><br>
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Diagonal symmetry</p>
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</div>
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</div>
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<p>In other words, there is a <b>mirror-image</b>.</p>
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<h2>Benefits</h2>
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<p>The benefits of finding symmetry in an equation are:</p>
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<ul>
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<li>we understand the equation better</li>
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<li>it is easier to plot</li>
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<li>it can be easier to solve. When we find a solution on one side, we can then say "also, by symmetry, the (mirrored value)"</li>
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</ul>
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<h2>How to Check For Symmetry</h2>
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<p>We can often see symmetry visually, but to be really sure we should check a simple fact:</p>
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<p class="center larger">Is the equation unchanged when using symmetric values?</p>
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<p>How we do this depends on the type of symmetry:</p>
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<h3>For Symmetry About Y-Axis</h3>
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<p>For symmetry with respect to the Y-Axis, check to see if the equation is the same when we replace <b>x</b> with <b><span class="larger">−</span>x</b>:</p>
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<div class="example">
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<h3>Example: is y = x<sup>2</sup> symmetric about the y-axis?</h3>
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<p>Try to replace <b>x</b> with <b><span class="larger">−</span>x</b>:</p>
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<p class="center larger">y = (−x)<sup>2</sup></p>
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<p>Since<span class="larger"> (−x)<sup>2</sup> = x<sup>2</sup></span> (<a href="../multiplying-negatives.html">multiplying a negative times a negative gives a positive</a>), there is no change</p>
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<p>So y = x<sup>2</sup> is symmetric about the y-axis</p>
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</div>
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<h3>For Symmetry About X-Axis</h3>
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<p>Use the same idea as for the Y-Axis, but try replacing <b>y</b> with <b><span class="larger">−</span>y</b>.</p>
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<div class="example">
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<h3>Example: is y = x<sup>3</sup> symmetric about the x-axis?</h3>
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<p>Try to replace <b>y</b> with <b><span class="larger">−</span>y</b>:</p>
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<p class="center larger">−y = x<sup>3</sup></p>
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<p>Now try to get the original equation:</p>
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<p>Try multiplying both sides by <span class="larger">−</span>1:</p>
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<p class="center"><span class="larger">y = −x<sup>3</sup></span></p>
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<p>It is different.</p>
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<p>So y = x<sup>3</sup> is <b>not</b> symmetric about the y-axis</p>
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</div>
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<p> </p>
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<h3>Diagonal Symmetry</h3>
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<p>Try swapping <b>y</b> and <b>x</b> (i.e. replace both <b>y</b> with <b>x</b> and <b>x</b> with <b>y)</b>.</p>
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<div class="example">
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<h3>Example: does y = 1/x have Diagonal Symmetry?</h3>
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<p>Start with:</p>
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<p class="center larger">y = 1/x</p>
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<p>Try swapping <b>y</b> with <b>x</b>:</p>
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<p class="center larger"><span class="hilite">x</span> = 1/<span class="hilite">y</span></p>
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<p> </p>
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<p>Now rearrange that: multiply both sides by <b>y</b>:</p>
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<p class="center larger">xy = 1</p>
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<p>Then divide both sides by <b>x</b>:</p>
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<p class="center larger">y = 1/x</p>
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<p>And we have the original equation. They are the same.</p>
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<p>So y = 1/x has Diagonal Symmetry</p>
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</div>
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<h3>Origin Symmetry</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../geometry/images/origin-symmetry.jpg" alt="origin symmetry" height="207" width="258" ></p>
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<p><a href="../geometry/symmetry-point.html">Origin Symmetry</a> is when every part has a matching part:</p>
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<ul>
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<li>the <b>same distance</b> from the central point</li>
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<li>but in the<b> opposite direction.</b></li>
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</ul>
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<p>Check to see if the equation is the same when we replace both <b>x</b> with <b><span class="larger">−</span>x</b> and <b>y</b> with <b><span class="larger">−</span>y</b>.</p>
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<div style="clear:both"></div>
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<div class="example">
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<h3>Example: does y = 1/x have Origin Symmetry?</h3>
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<p>Start with:</p>
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<p class="center larger">y = 1/x</p>
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<p>Replace x with <span class="larger">−</span>x and y with <span class="larger">−</span>y:</p>
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<p class="center larger">(<span class="hilite">−y</span>) = 1/(<span class="hilite">−x</span>)</p>
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<p>Multiply both sides by <span class="larger">−</span>1:</p>
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<p class="center larger">y = 1/x</p>
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<p>And we have the original equation.</p>
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<p>So y = 1/x has Origin Symmetry</p>
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</div>
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<p>Amazing! <span class="larger">y = 1/x</span> has origin symmetry as well as diagonal symmetry!</p>
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<p> </p>
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<div class="questions">1154, 1155, 2463, 2464, 2465, 4042, 4043, 4044, 4045, 4046</div>
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<div class="related">
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<a href="../geometry/symmetry.html">Symmetry</a>
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<a href="index.html">Algebra Index</a>
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