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146 lines
5.7 KiB
HTML
146 lines
5.7 KiB
HTML
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<title>Asymptote</title>
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<meta name="description" content="An asymptote is a line that a curve approaches, as it heads towards infinity" />
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Asymptote</h1>
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<p class="larger">An asymptote is a <b>line</b> that a curve approaches, as it heads towards infinity:</p>
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<p class="center"><img src="images/asymptote.svg" alt="Asymptote" /></p>
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<h2>Types</h2>
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<p>There are three types: horizontal, vertical and oblique:</p>
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<p class="center"><img src="images/asymptote-types.svg" alt="Asymptote Types" /></p>
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<p>The direction can also be negative:<br>
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<br>
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<img src="images/asymptote-neg-infinity.svg" alt="asymptote negative infinity" /></p>
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<p>The curve can approach from any side (such as from above or below for a horizontal asymptote),</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/asymptote-crossing.gif" alt="Asymptote Crossing" width="215" height="77" /></p>
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<p>or may actually cross over (possibly many times), and even move away and back again.</p>
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<div style="clear:both"></div><p>The important point is that:</p>
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<p>The <b>distance</b> between the curve and the asymptote <b>tends to zero</b> as they head to infinity (or −infinity)</p>
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</div>
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<h2>Horizontal Asymptotes</h2>
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<table border="0">
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<tr>
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<td><img src="images/asymptote-horizontal.svg" alt="Horizontal Asymptote" /></td>
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<td> </td>
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<td><p>It is a Horizontal Asymptote when:</p>
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<p>as x goes to infinity (or −infinity) the curve approaches some constant value <b>b</b></p></td>
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</tr>
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</table>
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<h2>Vertical Asymptotes</h2>
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<table border="0">
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<tr>
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<td><img src="images/asymptote-vertical.svg" alt="Vertical Asymptote" /></td>
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<td> </td>
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<td><p>It is a Vertical Asymptote when:</p>
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<p>as x approaches some constant value <b>c</b> (from the left or right) then the curve goes towards infinity (or −infinity).</p></td>
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</tr>
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</table>
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<h2>Oblique Asymptotes</h2>
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<table border="0">
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<tr>
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<td><img src="images/asymptote-oblique.svg" alt="Oblique Asymptote" /></td>
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<td> </td>
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<td><p>It is an Oblique Asymptote when:</p>
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<p>as x goes to infinity (or −infinity) then the curve goes towards a line <b>y=mx+b</b></p>
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<p>(note: m is not zero as that is a Horizontal Asymptote).</p></td>
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</tr>
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</table>
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<p> </p>
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<div class="example">
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<h3>Example: (x<sup>2</sup>−3x)/(2x−2)</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/asymptote-example.gif" alt="xy graph of (x^2-3x)/(2x-2)" width="191" height="193" /></p>
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<p> </p>
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<p>The <a href="../data/function-grapher090b.html?func1=(x^2-3x)/(2x-2)&func2=x/2-1&xmin=-10&xmax=10&ymin=-6.17&ymax=7.17">graph of (x<sup>2</sup>-3x)/(2x-2)</a> has:</p>
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<ul>
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<li>A vertical asymptote at <b>x=1</b></li>
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<li>An oblique asymptote: <b>y=x/2 − 1</b></li>
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</ul>
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</div>
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<p> </p>
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<p>These questions will only make sense when you know <a href="rational-expression.html">Rational Expressions</a>:</p>
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<div class="questions">
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<script>getQ(8432, 8434, 9666, 9667, 9668, 8433, 8435, 8436, 8437, 9669);</script>
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</div>
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<div class="related">
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<a href="rational-expression.html">Rational Expressions</a>
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<a href="index.html">Algebra Index</a>
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Copyright © 2020 MathsIsFun.com
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