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      <text>1.3.5 (Roots and curve sketching)</text>
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      <text><![CDATA[<h1>The importance of roots</h1>
<p>One of the most important mathematical skills is that of curve sketching. That is, being able to accurately plot a graph from a given equation. Whilst this is relatively straightforward for a straight line, it becomes a more complicated process for parabolas as a parabola has more factors governing its behaviour. In order to accurately sketch a parabola, we need to find the points at which it crosses both the \(x\) and \(y\) axes, its turning point and whether it is concave upwards or concave downwards. Fortunately we can obtain this information from any given quadratic and we will see precisely how to do this in the rest of this section.</p>
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<p>Recall that the <em>roots</em> of the quadratic function \(y = ax^2 + bx +c\) are the values of \(x\) such that \(y = 0\). So how does this correspond to the graph of a quadratic? Notice that the equation of the \(x\)-axis of a graph is also the equation \(y = 0\). Therefore the roots of the quadratic correspond to the points at which it crosses the \(x\)-axis. Similarly, the equation of the \(y\)-axis is \(x = 0\). So by inputing the value \(x = 0\) into the initial quadratic we obtain the \(y\)-intercept. The coefficient of \(x^2\) controls the width and shape of the graph.</p>
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<p>Recall that the nature of the roots of a quadratic are governed by the <em>discriminant</em> \(\Delta = b^2 - 4ac\). Specifically:</p>
<p>If the discriminant \(\Delta &gt; 0\): the quadratic has <em>two distinct real roots</em>,</p>
<p>If the discriminant \(\Delta = 0\): the quadratic has <em>one repeated real root</em>,</p>
<p>If the discriminant \(\Delta &lt; 0\): the quadratic has <em>no real roots.</em> <br /><br /></p>
<p>We can ask what this means graphically for our general quadratic. If \(\Delta &gt; 0\) then the parabola crosses the \(x\)-axis twice, and the \(x\)-coordinates of the points where it crosses are the roots of the quadratic. If \(\Delta = 0\) then the parabola touches the \(x\)-axis and the \(x\)-coordinate of the point at which it touches the \(x\)-axis is the repeated root of the quadratic. If \(\Delta &lt; 0\): the parabola does not cross the \(x\)-axis. We can also note that if \(a &gt; 0\) then the parabola is concave upwards, whereas if \(a &lt; 0\) the parabola is concave downwards. We can see all of these possibilities below.</p>
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<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Parabola%20possibilities.png" alt="Possible graphs of a parabola" width="648" height="355" /></p>]]></text>
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