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197 lines
7.1 KiB
HTML
197 lines
7.1 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/sets/graph-equation.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:02:21 GMT -->
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<head>
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<meta http-equiv="content-type" content="text/html; charset=UTF-8">
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<!-- #BeginEditable "doctitle" -->
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<title>Graph of an Equation</title>
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<meta name="Description" content="Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.">
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<!-- #EndEditable -->
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<meta name="keywords" content="math, maths, mathematics, school, homework, education">
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<meta name="HandheldFriendly" content="true">
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<meta name="referrer" content="always">
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<link rel="stylesheet" href="../style4.css">
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<script src="../main4.js"></script>
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<script>document.write(gTagHTML())</script>
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</head>
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<body id="bodybg">
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<div id="stt"></div>
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<div id="adTop"></div>
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<header>
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<div id="hdr"></div>
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<div id="tran"></div>
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<div id="adHide"></div>
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<div id="cookOK"></div>
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</header>
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<div class="mid">
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<nav>
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<div id="menuWide" class="menu"></div>
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<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
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<div id="search" role="search"></div>
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<div id="linkto"></div>
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<div id="menuSlim" class="menu"></div>
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<div id="menuTiny" class="menu"></div>
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</nav>
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<div id="extra"></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Function Graph</h1>
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<p class="center"><img src="../data/images/graph-func-gen.svg" alt="x y graph"><br>An example of a function graph</p>
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<h2>How to Draw a Function Graph
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</h2>
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<p>First, start with a blank graph like this. It has x-values going left-to-right, and y-values going bottom-to-top:</p>
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<p class="center"><img src="../data/images/graph-gen.svg" alt="x y graph"><br>
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The x-axis and y-axis cross over <br>where x and y are both zero.</p>
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<h2>Plotting Points</h2>
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<p>A simple (but not perfect) approach is to calculate the function at <b>some points</b> and then plot them.</p>
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<p>A function graph is the <a href="set-of-points.html">set of points</a> of the values taken by the function.</p>
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<p></p>
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<div class="example">
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<h3>Example: <span class="large">y = x<sup>2</sup> − 5</span></h3>
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<p>Let us calculate <b>some points</b>:</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 style="text-align:center;">
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<th width="90">x</th>
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<th>y = x<sup>2</sup>−5</th>
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</tr>
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<tr style="text-align:center;">
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<td>−2</td>
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<td>−1</td>
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</tr>
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<tr style="text-align:center;">
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<td>0</td>
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<td>−5</td>
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</tr>
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<tr style="text-align:center;">
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<td>1</td>
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<td>−4</td>
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</tr>
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<tr style="text-align:center;">
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<td>3</td>
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<td>4</td>
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</tr>
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</tbody></table>
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</div>
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<p>And plot them like this:</p>
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<p class="center"><img src="../data/images/graph-x2m5a.svg" alt="x y graph"><br></p>
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<p>Not very helpful yet. Let us add some <b>more points</b>:</p>
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<p class="center"><img src="../data/images/graph-x2m5b.svg" alt="x y graph"></p>
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<p>Looking better!</p>
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<p>We can now guess that plotting <b>all the points</b> will look like this:</p>
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<p class="center"><img src="../data/images/graph-x2m5c.svg" alt="x y graph"></p>
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<p>A nice <a href="../geometry/parabola.html">parabola</a>.</p>
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</div>
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<p> </p>
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<p>We should try to plot enough points to be confident in what is going on!</p>
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<div class="example">
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<h3>Example: <span class="large">y = x<sup>3</sup> − 5x</span></h3>
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<p>With these calculated points:</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 style="text-align:center;">
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<th width="90">x</th>
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<th>y = x<sup>3</sup>−5x</th>
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</tr>
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<tr style="text-align:center;">
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<td>−2</td>
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<td>2</td>
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</tr>
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<tr style="text-align:center;">
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<td>0</td>
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<td>0</td>
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</tr>
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<tr style="text-align:center;">
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<td>2</td>
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<td>−2</td>
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</tr>
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</tbody></table>
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</div><br>
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<p>We might think this is the graph:</p>
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<p class="center"><img src="images/graph-x3-5-a.gif" alt="graph" height="201" width="201"></p>
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<p>But this is the real graph:</p>
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<p class="center"><img src="images/graph-x3-5-b.gif" alt="graph" height="199" width="199"></p>
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</div>So "plotting some points" is useful, but <b>can lead to mistakes</b>.
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<h2>Complete Graph</h2>
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<p>For a graph to be "complete" we need to show all the important features:</p>
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<ul>
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<li>Crossing points</li>
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<li>Peaks</li>
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<li>Valleys</li>
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<li>Flat areas</li>
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<li><a href="../algebra/asymptote.html">Asymptotes</a></li>
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<li>Any other special features</li></ul>
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<p>This often means thinking carefully about the function.</p>
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<div class="example">
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<h3>Example: (x−1)/(x<sup>2</sup>−9) </h3>
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<p>On the page <a href="../algebra/rational-expression.html">Rational Expressions</a> we do some work to discover that the function:</p>
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<ul>
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<li>crosses the x-axis at 1, </li>
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<li>crosses the y-axis at 1/9,</li>
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<li>has vertical asymptotes (where it heads towards minus/plus infinity) at −3 and +3</li></ul>
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<p>The result is that we can make this sketch:</p>
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<p class="center"><img src="../algebra/images/asymptote-example2b.svg" alt="function local minimum and maximum"><br>
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Sketch of (x−1)/(x<sup>2</sup>−9) from <a href="../algebra/rational-expression.html">Rational Expressions</a>.</p>
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</div>
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<h2>Using Calculus</h2>
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<p>We can also <span class="center"><a href="../calculus/maxima-minima.html">find Maxima and Minima using derivatives</a></span> :</p>
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<p class="center"><img src="../calculus/images/function-min-max.svg" alt="function local minimum and maximum"><br></p>
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<h2>Tools to Help You
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</h2>
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<div class="bigul">
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<ul>
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<li>The <a href="../data/function-grapher.html">Function Grapher</a> can help you. Enter the equation as "y=(some function of x)". You can use zoom to find important points.</li>
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<li>If you can't write the equation as "y=(some function of x)", you can try the <a href="../data/grapher-equation.html">Equation Grapher</a>, where you enter equations like "x^2+y^2=9" (meaning <b>x<sup>2</sup>+y<sup>2</sup>=9</b>).</li></ul></div>
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<p>But remember they are just a help! They are only computer programs,
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and could easily miss some important thing on the graph, or not plot
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something correctly.</p>
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<p><br></p>
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<div class="words">
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<p>Note: you may hear the phrase "satisfy the equation", which means where the equation is <b>true</b>.</p></div>
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<p></p>
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<p><br></p>
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<div class="questions">
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<script>getQ(4036, 1150, 1151, 4037, 2165, 7275, 7276, 7277, 7278, 7279);</script>
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</div>
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<div class="related">
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<a href="../algebra/index.html">Algebra Index</a>
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<a href="../data/index.html">Data Index</a>
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</div>
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<!-- #EndEditable -->
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</article>
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<div id="adend" class="centerfull noprint"></div>
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<footer id="footer" class="centerfull noprint"></footer>
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<div id="copyrt">Copyright © 2020 MathsIsFun.com</div>
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</body><!-- #EndTemplate -->
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