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<p>To reinforce the connections between the parameters of a quadratic expression and the situation it describes, ask students:</p>
<ul>
  <li> &ldquo;So far, we&rsquo;ve seen different expressions that represent vertical distances. Here are three expressions that all represent distance, in feet, as a function of time, in seconds, for an object that is dropped or launched. What does each of them tell us? Draw a diagram to illustrate the distances, if helpful.&rdquo; </li>
  <ul>
    <li> \(16t^2\) (The distance an object travels \(t\) seconds after being dropped.) </li>
    <li> \(400-16t^2\) (The height of an object that is dropped from a height of 400 feet.) </li>
    <li> \(50+100t-16t^2\) (The height of an object that is shot up from 50 feet above the ground at a vertical speed of 100 feet per second, \(t\) seconds after being launched.) </li>
  </ul>
  <li> &ldquo;If each expression defines a function, what does the zero of that function tell us?&rdquo; </li>
  <ul>
    <li> \(16t^2\) (The zero is the time when the object has traveled a distance of 0 feet. This happens at \(t=0\), before the object is dropped.) </li>
    <li> \(400-16t^2\) (The zero is the time when the height of the object is 0 feet, which is when it hits the ground.) </li>
    <li> \(50+100t-16t^2\) (The zero is the time when the height of the object is 0 feet, which is also when it hits the ground.) </li>
  </ul>
</ul>
<p>Explain to students that the models seen here are simplified models, and they ignore other factors, such as air resistance. The models that scientists use to study physical phenomena are likely to be more complex than what students have seen here.</p>
<p>If time permits, consider addressing a common misconception: that a graph of a quadratic function that represents distance-time relationship shows the physical trajectory of the object. Ask students to draw a sketch of what a bystander would see if they are facing the cannon as the ball is being launched. If needed, revisit the GeoGebra applet &ldquo;Distance as a Quadratic Function of Elapsed Time&rdquo; (from the previous lesson) to further emphasize this difference.</p>
<p>Clarify that the graph represents the height of the object as a function of time, not the path that the object travels. In the examples given here, the object just goes straight up and straight down.</p>