86fce053-1160-4d80-9505-bf683df54660.html

<h3>Activity</h3>

<p>TThis activity is an extension of the expectations in the TEKS.</p>

<p>This activity further addresses the skill of choosing an appropriate graphing window when using graphing technology. In an earlier activity, students looked at how adjusting the graphing window affects the usefulness of the graph. Here, they gauge the reasonableness of a graphing window given an equation and a description of a function.</p>
<p>Because exponential functions eventually grow very rapidly, the graph tends to quickly go off the screen if the domain is too large. The graphing window can be adjusted to display large values for the vertical axis, but in doing so, the output values for most of the domain will all look like they are essentially 0.</p>
<p>To decide on the reasonableness of the given graphing boundaries, students may evaluate the function at the endpoints of the domain for the graphing window. Look for students who think carefully about the domain, observing that, based on the context, the equation probably does not model even modest negative values of the input variable. This activity represents scaffolded practice for an important aspect of mathematical modeling.</p>
<h4>Launch</h4>
<p>Provide access to graphing technology.</p>
<h4>Student Activity</h4>
<p>The equation \(m=20 \cdot (0.8)^h\) models the amount of medicine \(m\) (in milligrams) in a patient&rsquo;s body as a function of hours, \(h\), after injection.</p>
<ol class="os-raise-noindent">
  <li> Without using a graphing tool, decide if the following horizontal and vertical boundaries are suitable for graphing this function.&nbsp; <br>
    <br>
    Be prepared to show your reasoning.<br>
    \(-10&lt;h&lt;100\)<br>
    \(
    -100&lt;m&lt;1,000\) <br>
    <br>
    <strong>Answer:</strong> Compare your answer:<br>
    <br>
    <p>No, these boundaries are not suitable for graphing this function.</p>
    <ul>
    <li>Horizontal axis: There is probably no medicine left in the body 100 hours after injection, so it is not helpful to graph all the way to an \(h\) value of 100. And if there was any medicine in the body before injection, it wouldn’t be accounted for by this equation, so \(h\) should probably start at 0. 
    </li>
    <li>Vertical axis: The medicine is decreasing in amount, starting from 20 milligrams, so 1,000 for the upper end of the axis is far too large. The negative values for 𝑚 are also not needed because a negative amount of medicine would not make any sense here.
  </li>
</ul>


</ol>
<ol class="os-raise-noindent" start="2">
  <li> Verify your answer by graphing the equation using graphing technology and using the given graphing window. What do you see? Sketch or describe the graph. <br>
    <br>
    <strong>Answer:</strong> Compare your answer: <br>
    <br>
    <img alt class="img-fluid atto_image_button_text-bottom" height="272" role="presentation" src="https://k12.openstax.org/contents/raise/resources/f6d2bee0fee13f3c5de59625520542f4a72443a0" width="280">
  </li>
</ol>
<ol class="os-raise-noindent" start="3">
  <li>If your graph in the previous question is unhelpful, modify the window settings so that the graph is more useful. Record the window settings here. Convince a partner why the horizontal and vertical boundaries that you set are better. <br>
    <br>
    <strong>Answer:</strong> Compare your answer: <br>
    <br>
    Your answer may vary, but here are some samples: A reasonable graph would have an \(h\)-axis that shows values from 0 to around 20 and an \(m\)-axis showing values from 0 to somewhere in the low 20s. Here is an example :<br>
    <br>
    Points plotted on the coordinate plane, showing exponential decay. Horizontal axis from 0 to 22.5, labeled \(h\). Vertical axis from 0 to 22.5, labeled \(m\). <br>
    <br>
    <img height="289" src="https://k12.openstax.org/contents/raise/resources/2649f4f4494087f8193b76552c86d167944a3c45" width="283">
  </li>
</ol>