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      <text>4.3.2 (Plotting exponential example)</text>
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      <text><![CDATA[<h3>Plotting exponential functions</h3>
<p>In general, plotting exponential functions can be more difficult than plotting polynomials. However, there are certain properties we can use in order to make this more straightforward. Suppose that we wish to plot a general exponential function \(f(x) = ka^{x+b} + c\). First of all, as we can see using the applet above, note that all exponential functions follow a similar shape, so in order to accurately sketch such graphs we need to find the following information:</p>
<ol>
<li>What happens to the function \(f(x)\) as \(x \to \pm \infty\)?</li>
<li>What is the \(y\)-intercept of the graph?</li>
<li>We need two different points to draw the curve.</li>
</ol>
<p>We shall see this by way of the following example.</p>
<p> </p>
<h3>Example</h3>
<p>Sketch the graph of \(y = 2^{-x} - 3\) and therefore state the domain and range of the function.</p>
<h3>Solution</h3>
<p>First, notice that as \(x \to \infty\) then \(2^{-x} = \frac{1}{2^x} \to 0\), therefore \(y \to -3\). Therefore the graph tends towards the horizontal line \(y = -3\). This is sometimes called the <b>horizontal asymptote</b> of the function.</p>
<p>Similarly, as \(x \to -\infty\), \(2^{-x} \to \infty\) therefore \(y \to \infty\).</p>
<p>The \(y\)-intercept is the value of \(y\) when we substitute \(y =0\) into the given equation, therefore \[ y = 2^0 - 3 = 1- 3 = -2. \] So the \(y\)-intercept is \(-2\).</p>
<p>Now let us find two points on the curve, let us input \(x = -2\) and \(x = -2\) to find their corresponding y-values. \[ x = 2 \implies y = 2^{-2} - 3 = \frac{1}{4}-3 = -2\frac{3}{4}, \] similarly, \[ x=-2 \implies y = =2^2 - 3 = 1. \] Thus two points on the curve are \((2, -2\frac{3}{4})\) and \((-2,1)\).</p>
<p>Therefore we can put all of this information to sketch the following graph.</p>
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<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Exponential%20function.png" alt="graph of exponential function" width="700" height="421" /></p>
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<p>We can see that the domain of \(f(x)\) is \(\{x : x\in\mathbb{R}\}\) and the range is \(\{y : y &gt; 3\}\).</p>]]></text>
<file name="Exponential function.png" path="/" encoding="base64">[encoded data, 93728 characters]</file>
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