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<h3>Warm Up (5 minutes)</h3>
<p>In this warm up, students begin to apply their new understandings about graphs to reason about quadratic functions contextually. They evaluate a simple quadratic function, find its maximum, and interpret these values in context. The input values to be evaluated produce an output of 0, reminding students of the meaning of the zeros of a function and their connection to the horizontal intercepts (\(x\)-intercepts) of the graph.</p>
<p>To find the maximum value of the function, students could graph the function, but applying what they learned about the connection between the horizontal intercepts and the vertex (without graphing) would be more efficient. Identify students who make this connection and ask them to share their thinking in the discussion.</p>
<h4>Launch</h4>
<p>Give students access to a calculator.</p>
<h4>Student Activity</h4>
<p>The height, in inches, of a frog's jump is modeled by the equation \(h(t)=60t-75t^2\), where the time \(t\), after the frog jumps, is measured in seconds.</p>
<p><img alt="close-up of a frog on plant stem" height="318" src="https://k12.openstax.org/contents/raise/resources/dc7c6fde68fd9c5493e88f22a1997baa7fa2c8c5" width="478"></p>
<ol class="os-raise-noindent">
  <li>For \(h(t)=60t-75t^2\):</li>
</ol>
<ol class="os-raise-noindent">
  <ol class="os-raise-noindent" type="a">
    <li> Find \(h(0)\). </li>
  </ol>
  <p><strong>Answer:</strong> 0</p>
</ol>
<ol class="os-raise-noindent">
  <ol class="os-raise-noindent" start="2" type="a">
    <li> Find \(h(0.8)\). </li>
  </ol>
  <p><strong>Answer:</strong> 0</p>
</ol>
<ol class="os-raise-noindent" start="2">
  <li>What do these values of \(h(0)\) and \(h(0.8)\) mean in terms of the frog's jump?</li>
</ol>
<p><strong>Answer: </strong>They mean that the frog is on the ground when \(t=0\) (before jumping) and when \(t=0.8\) (when the frog lands on the ground).</p>
<ol class="os-raise-noindent" start="3">
  <li>After it jumped, how much time passed, in seconds, before the frog reached its maximum height?</li>
</ol>
<p><strong>Answer:</strong> 0.4 seconds</p>
<ol class="os-raise-noindent" start="4">
  <li>Explain how you know this is the time the frog reached the maximum height.

  </li>
</ol>
<p><strong>Answer:</strong> The frog reached its maximum height at \(t=0.4\), as that is the halfway point of the jump. If we graph the function, the vertex of the graph will have a horizontal coordinate (\(t\)-coordinate) of 0.4.</p>
<h4>Activity Synthesis</h4>
<p>Select students to share their responses and explanations. Even though students are not asked to graph the function, make sure that they begin to connect the quadratic expression that defines the function to the features of the graph representing that function.</p>
<p>Ask students: "If we graph the equation, what would the graph look like? Where would the intercepts be? Would the graph open up or down? Where would the vertex be?"</p>
<p>Highlight the following points:</p>
<ul>
  <li> The graph intersects the horizontal axis at \(t=0\) and \(t=0.8\), because \(h(0)\) and \(h(0.8)\) both have a value of 0. </li>
  <li> \(h(0.4)\) is the maximum height of the frog since \(t=0.4\) is halfway between the horizontal intercepts. If we were to graph it, (\(0.4,h(0.4)\)) would be the vertex. </li>
  <li> The graph opens downward because the coefficient of the squared term is a negative number (-75). </li>
</ul>
<p><img alt="Graph of parabola" height="213" src="https://k12.openstax.org/contents/raise/resources/3723775573022397624cf1f5b199a0de2ae3ba6f" width="315"></p>