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    <name>
      <text>2.7.2.4 Study the graph of a rational function 2</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Consider the graph and the algebraic form of the rational function \(R_3(z)= \dfrac{2z^2+z-1}{(z-1)(z+2)}\).</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/2.7.2.3%20R_3%28z%29.svg" alt="" role="presentation" class="img-responsive atto_image_button_text-bottom" width="356" height="230">
</figure>
<ol class="HELM_parts">
<li>What happens to the function when \(z=-2\) and \(z=1\)? [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]<br></li>
<li>Which values of \(z\) should be excluded from the domain of this function? [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</li>
<li>Substitute some values for \(z\) (e.g. \(10, 100...\)). What happens to \(R_3\) as \(z\) becomes larger and larger (namely, as \(z\) goes to infinity)? [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</li>
<li>We can observe that the function \(R_3(z)\) has an horizontal asymptote. The equation of this asymptote is: \(y=\)[[input:ans4]] [[validation:ans4]] [[feedback:prt4]]</li>
<li>What is the name given to the lines \(z=1\) and \(z=−2\)? [[input:ans5]] [[validation:ans5]] [[feedback:prt5]]</li>
</ol>]]></text>
<file name="2.7.2.3 R_3(z).svg" path="/" encoding="base64">[encoded data, 22240 characters]</file>
<file name="block8-fig4-1.svg" path="/" encoding="base64">[encoded data, 22240 characters]</file>
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      <text><![CDATA[<ol class="HELM_parts">
<li>When \(z=-2\) and \(z=1\) the value of the function blows up to infinity because the denominator goes to zero (if we substitute \(z=-2\), for example, in \((z-1)(z+2)=(-2-1)(-2+2)=-3*0=0\)) . To be more precise, when we approach \(z=-2\) from the left and \(z=1\) from the right, the function approaches +∞. If instead we approach  \(z=-2\) and \(z=1\) from the right and from the left respectively, the value of the function goes to \(- \infty\).</li>
<li>The points \(z=-2\) and \(z=1\) are not in the domain since they represent singularities for the function (the denominator of the function is zero in these cases so the value of the function explodes).</li>
<li>If you look at the graph you can notice that for positive and negative large values of \(z\) (\(z \to \pm \infty\)), the value of the function tends at a constant value.</li>
<li>The constant value of the previous item is \(y=2\). This line is called to be an horizontal asymptot for the function.</li>
<li>The lines \(z=1\) and \(z=−2\) are called vertical asymptotes.</li></ol>]]></text>
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      <text>2020120600</text>
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      <text><![CDATA[ta1: [["The denominator goes to infinity", false], ["The function goes to zero", false], ["The denominator is zero, the function goes to infinity", true], ["The numerator tends to infinity", false]];
ta2:[["z=-1, z=2", false],["z=-2, z=1", true],["z=-2, z=-1", false],["z=1 only", false]];
ta3: [["The function goes to infinity", false],["The function goes to zero", false] ,["The function approaches the value -1", false], ["The function approaches the value 2", true]];
correct4: 2;
ta5:[["Oblique asymptotes", false], ["Vertical asymptotes", true], ["Horizontal asymptotes", false], ["Vertical lines", false]];]]></text>
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    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
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