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      <text>2.5.1.1 Linear functions</text>
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      <text><![CDATA[<h3> Linear functions </h3>


<p>Any function of the form <span class="math inline">\(y=f(x)=ax+b\)</span>, where <span class="math inline">a</span> and <span class="math inline">b</span> are constants, is called a <b>linear</b> <b>function</b>. The constant \(a\) is called the <b>coefficient </b>of \(x\), and \(b\) is referred to as the <b>constant term</b>.<br>
</p><div class="HELM_keypoint">
<h4>Key Point</h4>
All linear functions can be written in the form: <span class="math display">\[\qquad f(x)=ax+b\]</span>
where <i>a </i>and<i> b</i> are constants.
</div><br>
<p>For example, \(f(x)=3x+2\), \(g(x)=\dfrac12 x-7\), \(h(x)=-3x+\dfrac23\) and \(k(x)=2x\) are all linear functions.The graph of a linear function is always a straight line. Such a graph can be plotted by finding just two distinct points and joining these with a straight line.</p>

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<h4 class="HELM_example">Example</h4>
<p>Plot the graph of the linear function <span class="math inline"> \(y=f(x)=4x+3\) </span>.</p>
<h4 class="HELM_solution">Solution</h4>
<p>We start by finding two points. First we choose <span class="math inline"> \(x=0\) </span>, so <span class="math inline"> \(y=f(0)=3\) and</span> the first point has coordinates <span class="math inline"> \((0, 3)\) </span>. Secondly, suppose we choose <span class="math inline"> \(x= 5\)</span>, then <span class="math inline"> \(y=f(5) = 23\) </span>. The second point is <span class="math inline"> \((5,23)\)</span>. These two points are plotted and then joined by a straight line as shown in the following diagram. <br></p>
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<img src="@@PLUGINFILE@@/2.5.1.1_linear_function.svg" alt="" role="presentation" class="img-responsive atto_image_button_text-bottom" width="201" height="169">
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