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      <text>1.3.1 (Quadratic functions)</text>
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      <text><![CDATA[<h1>Quadratic Functions</h1>
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<p>We are familiar with the notion of a linear function. This is a function which has the form \(y = mx + c\), where \(m\) corresponds to the gradient and \(c\) is the \(y\)-intercept. Graphs of these equation correspond to straight lines.</p>
<p>In contrast, a <strong>quadratic function</strong> is an equation of the form \(y = ax^2 + bx + c\), where \(a \not= 0\). This allows us to plot a quadratic in the Cartesian plane i.e. it allows us to plot quadratics on an \(x-y\) axis. Note that sometimes the notation \(f(x) = ax^2 + bx + c\) is used to show explicitly that it is a function of \(x\), we will discuss this in more detail in a later week. The squared term of the quadratic ensures that the resulting graph is no longer a straight line but is a curve called a <strong>parabola</strong>. This is an extremely important type of curve as they appear a lot in everyday life, such as in the curve of suspension bridges or the curve water makes when flowing out of a fountain. We shall begin by introducing some terminology.</p>
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<h1><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Parabola%20terminology.png" alt="Labelled graph of a parabola" width="424" height="424" /></h1>
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<li>The point at which the graph 'turns' is called the <strong>vertex</strong> or the <strong>turning point</strong>.</li>
<li>The turning point is a <strong>minimum</strong> if the graph curves upwards (is <strong>concave upwards</strong>).</li>
<li>The turning point is a <strong>maximum</strong> if the graph curves downwards (is <strong>concave downwards</strong>).</li>
<li>The vertical line which goes through the vertex is called the <strong>axis of symmetry</strong> and every parabola is symmetrical about its axis of symmetry.</li>
<li>The value of \(y\) where the graph crosses the y-axis is called the <strong>\(y\)-intercept</strong>.</li>
<li>The values of \(x\) where the graph crosses the \(x\)-axis are called the <b>\(x\)-intercepts</b> and they correspond to the roots of \(y = ax^2 + bx + c = 0\).</li>
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