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<h3>Warm Up (5 minutes)</h3>
<p>In this warm up, students revisit quadratic expressions in different forms and what the expressions reveal about the graphs that represent them. They match equations and graphs representing two quadratic functions. The axes of the graphs are unlabeled, so students need to reason abstractly about the parameters in the expressions and the features of the graphs.</p>
<h4>Launch</h4>
<p>Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down.&nbsp;</p>
<h4>Student Activity</h4>
<br>
<p><img height="182" src="https://k12.openstax.org/contents/raise/resources/05c4d743874e6414e9d720787af98245ec2791ad" width="254"></p>
<p>Here are graphs representing the functions \(f\) and \(g\) given by \(f(x)=x(x+6)\) and \(g(x)=x(x+6)+4\).</p>
<ol class="os-raise-noindent">
  <li>Which graph represents \(f(x)=x(x+6)\)? How do you know?</li>
</ol>
<p>Write down your answer, then select the<strong> solution</strong> button to compare your work.</p>
<p><strong>Answer: </strong>The lower graph (which goes through the origin) is the graph of \(f\).&nbsp;</p>
<ul>
  <li> The value of \(g\) is always 4 greater than the value of \(f\), no matter which value of \(x\) we choose. </li>
  <li> Rewriting the equation in standard form gives \(f(x)=x^2+6x\). This form shows that the \(y\)-intercept of the graph representing \(f\) is \((0,0)\). </li>
  <li> The expression that defines function \(f\) is in factored form. It shows that the two \(x\)-intercepts are \((0,0)\) and \((-6,0)\). </li>
</ul>
<ol class="os-raise-noindent" start="2">
  <li>Which graph represents \(g(x)=x(x+6)+4\)? How do you know?</li>
</ol>
<p>Write down your answer, then select the <strong>solution </strong>button to compare your work.</p>
<p><strong>Answer: </strong>The lower graph (which goes through the origin) is the graph of \(f\) and the upper one is the graph of \(g\).&nbsp;&nbsp;</p>
<ul>
  <li> The value of \(g\) is always 4 greater than the value of \(f\), no matter which value of \(x\) we choose. </li>
  <li> Rewriting the equation in standard form gives \(g(x)=x^2+6x+4\).This form shows that the \(y\)-intercept of the graph representing \(g\) is \((0,4)\). </li>
</ul>
<ol class="os-raise-noindent" start="3">
  <li>Where does the graph of \(f\) meet the \(x\)-axis? Explain how you know.</li>
</ol>
<p><strong>Answer: </strong>\(x=-6\) and \(x=0\), because these are the two values of \(x\) that make \(x(x+6)\)&nbsp; equal to 0.</p>
<h4>Activity Synthesis</h4>
<p>Invite students to share their matches and explanations. If one or more explanations in the student response are not mentioned, bring them up so that students can see multiple ways of reasoning about the equations and graphs.</p>