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<title>Graphing Quadratic Equations</title>
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Graphing Quadratic Equations</h1>
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<p class="center"><img src="images/quadratic-equation.svg" alt="Quadratic Equation"><br>
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A <a href="quadratic-equation.html">Quadratic Equation</a> in Standard Form<br>
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(<b>a</b>, <b>b</b>, and <b>c</b> can have any value, except that <b>a</b> can't be 0.)</p>
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<p>Here is an example:</p>
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<p class="center"><img src="images/quadratic-equation-reason.svg" alt="Quadratic Equation"></p>
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<h2>Graphing</h2>
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<p>You can graph a Quadratic Equation using the <a href="../data/function-grapher.html">Function Grapher</a>, but to <b>really understand</b> what is going on, you can make the graph yourself. Read On!</p>
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<h2>The Simplest Quadratic</h2>
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<p>The simplest Quadratic Equation is:</p>
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<p class="largest" align="center">f(x) = x<sup>2</sup></p>
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<p>And its graph is simple too:</p>
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<p class="center"><img src="../sets/images/function-square.svg" alt="Square function"></p>
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<p class="center"><span class="larger">This is the curve f(x) = x<sup>2</sup></span><br>
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It is a <a href="../geometry/parabola.html">parabola</a>.</p>
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<p>Now let us see what happens when we introduce the <span class="large">"a"</span> value:</p>
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<p class="center"><span class="largest">f(x) = ax<sup>2</sup></span></p>
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<p class="center"><img src="images/graph-ax2.gif" alt="ax^2" height="193" width="177"></p>
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<ul>
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<li>Larger values of <b>a</b> squash the curve inwards</li>
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<li>Smaller values of <b>a</b> expand it outwards</li>
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<li>And negative values of <b>a</b> flip it upside down</li>
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</ul>
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<p class="center"> </p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td>
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<a href="quadratic-equation-graph.html"><img src="images/quadratic-graph.svg" alt="Quadratic Graph"></a>
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</td>
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<td> </td>
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<td>
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<h2>Play With It</h2>
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<p>Now is a good time to play with the<br>
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"<a href="quadratic-equation-graph.html">Quadratic Equation Explorer</a>" so you can<br>
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see what different values of <b>a</b>, <b>b</b> and <b>c</b> do.</p>
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</td>
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</tr>
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</tbody></table>
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<h2>The "General" Quadratic</h2>
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<p>Before graphing we <b>rearrange</b> the equation, from this:</p>
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<p class="center"><span class="largest">f(x) = ax<sup>2</sup> + bx + c</span></p>
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<p>To this:</p>
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<p class="center"><span class="largest">f(x) = a(x-h)<sup>2</sup> + k</span></p>
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<p>Where:</p>
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<ul>
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<div class="bigul">
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<li>h = −b/2a</li>
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<li>k = <i><b>f( </b></i>h <i><b>)</b></i></li>
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</div>
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</ul>
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<p>In other words, calculate <b>h</b> (= −b/2a), then find <b>k</b> by calculating the whole equation for <b>x=h</b></p>
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<h2>But Why?</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/quadratic-vertex.svg" alt="quadratic vertex"></p>
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<p>The wonderful thing about this new form is that <span class="large">h</span> and <span class="large">k</span> show us the very lowest (or very highest) point, called the <b>vertex</b>:</p>
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<p>And also the curve is <a href="../geometry/symmetry-reflection.html">symmetrical</a> (mirror image) about the <b>axis</b> that passes through <b>x=h</b>, making it easy to graph</p>
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<p> </p>
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<h3>So ...</h3>
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<ul>
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<li><b>h</b> shows us how far left (or right) the curve has been shifted from x=0</li>
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<li><b>k</b> shows us how far up (or down) the curve has been shifted from y=0</li>
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</ul>
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<p>Lets see an example of how to do this:</p>
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<div class="example">
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<h3>Example: Plot f(x) = 2x<sup>2</sup> − 12x + 16</h3>
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<div class="example2"></div>
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<p>First, let's note down:</p>
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<ul>
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<li><b>a = 2, </b></li>
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<li><b>b = −12, </b>and</li>
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<li><b>c = 16</b></li>
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</ul>
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<p>Now, what do we know?</p>
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<ul>
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<li>a is positive, so it is an "upwards" graph ("U" shaped)</li>
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<li>a is 2, so it is a little "squashed" compared to the <b>x<sup>2 </sup></b>graph</li>
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</ul>
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<p>Next, let's calculate h:</p>
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<div class="so">h = −b/2a = −(−12)/(2x2) = <b>3</b></div>
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<p>And next we can calculate k (using h=3):</p>
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<div class="so">k = <i><b>f(</b></i>3<i><b>)</b></i> = 2(3)<sup>2</sup> − 12·3 + 16 = 18−36+16 = <b>−2</b></div>
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<p>So now we can plot the graph (with real understanding!):</p>
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<p class="center"><img src="images/graph-2x2m12xp16.gif" alt="2x^2-12x+16" height="163" width="515"></p>
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<p class="center">We also know: the <b>vertex</b> is (3,−2), and the <b>axis</b> is x=3</p>
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<div class="example2"></div>
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<div class="example2"></div>
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</div>
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<h2>From A Graph to The Equation</h2>
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<p>What if we have a graph, and want to find an equation?</p>
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<div class="example">
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<h3>Example: you have just plotted some interesting data, and it looks Quadratic:</h3>
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<p class="center"><img src="images/graph-quadratic-data.gif" alt="quadratic data" height="122" width="181"></p>
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<p>Just knowing those two points we can come up with an equation.</p>
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<p>Firstly, we know <b>h</b> and <b>k</b> (at the vertex):</p>
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<p class="center larger">(h, k) = (1, 1)</p>
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<p>So let's put that into this form of the equation:</p>
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<p class="center"><span class="larger">f(x) = a(x-h)<sup>2</sup> + k</span></p>
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<p class="center"><span class="larger">f(x) = a(x−1)<sup>2</sup> + 1</span></p>
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<p>Then we calculate "a":</p>
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<div class="tbl">
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<div class="row"><span class="left">We know the point <b>(0, 1.5)</b> so:</span><span class="right">f(0) = 1.5</span></div>
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<div class="row"><span class="left">And <b>a(x−1)<sup>2</sup> + 1</b> at x=0 is:</span><span class="right">f(0) = a(0−1)<sup>2</sup> + 1</span></div>
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<div class="row"><span class="left">They are both <b>f(0)</b> so make them equal:</span><span class="right"> a(0−1)<sup>2</sup> + 1 = 1.5</span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right">a + 1 = 1.5</span></div>
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<div class="row"><span class="left"> </span><span class="right">a = 0.5</span></div>
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</div>
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<div class="example2"></div>
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<p>And so here is the resulting Quadratic Equation:</p>
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<p class="center"><span class="large">f(x) = 0.5(x−1)<sup>2</sup> + 1</span></p>
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<p> </p>
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<p>Note: This may not be the <b>correct</b> equation for the data, but it’s a good model and the best we can come up with.</p>
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</div>
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<p> </p>
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<div class="questions">
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<script>getQ(567,7335,7336,7337,7338,1207,2443,2444,2445,2447);
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</script> </div>
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<div class="related">
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<a href="quadratic-equation.html">Quadratic Equations</a>
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<a href="factoring-quadratics.html">Factoring Quadratics</a>
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<a href="completing-square.html">Completing the Square</a>
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<a href="quadratic-equation-real-world.html">Real World Examples of Quadratic Equations</a>
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<a href="quadratic-equation-derivation.html">Derivation of Quadratic Equation Formula</a>
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<a href="index.html">Algebra Index</a>
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