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<h4>Activity (15 minutes)</h4>
<p>Previously, students were presented with descriptions of functions and, in one case, an equation that represents a function. In this activity, students are given the graph of a function and asked to analyze the underlying relationship and describe it using function notation.</p>
<p>The data used in this task are approximate, but the costs of solar energy during the time period mentioned in this activity can be appropriately modeled by an exponential decay model.</p>
<h4> Launch</h4>
<p>Elicit what students already know about solar cells. If students are unfamiliar with solar power and units of measurement for energy, give a brief introduction. Solar cells turn energy from the sun into electricity, a form of energy that is useful to humans. Energy is measured using units called watts. Over the course of the past several decades, the cost of solar cells has decreased. That is, we can manufacture solar cells that generate the same amount of electricity for less money.</p>
<p>Read the opening sentence of the task and display the graph for all to see. To help them process the information in the graph and think about functions, ask if the price per watt is a function of time and if the time is a function of the price per watt.</p>
<div class="os-raise-extrasupport">
  <div class="os-raise-extrasupport-header">
    <p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
    <p class="os-raise-extrasupport-name">Representation: Internalize Comprehension </p>
  <div class="os-raise-extrasupport-body">
    <p>Demonstrate and encourage students to use color coding and annotation to highlight connections between representations in a problem, for example, highlighting \(f(4)\) as written in the question and then highlighting the corresponding output value on the graph. Some students may benefit from reviewing that the \(x\)-coordinate is 4 and \(f(4)\) is the \(y\)-coordinate.</p>
    <p class="os-raise-text-italicize">Supports accessibility for: Visual-spatial processing</p>
  </div>
</div>
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<h4>Student Activity</h4>
<p>The cost, in dollars, to produce 1 watt of solar energy is a function of the number of years since 1977, \(t\).</p>
<p>From 1977 to 1987, the cost could be modeled by an exponential function \(f\). Here is the graph of the function.</p>
<p><img alt="A sheet of solar panels." src="https://k12.openstax.org/contents/raise/resources/fc18f868b9d9768cc0ff232f4b608bfb26081391" width="300"></p>
<p><img alt="Graph of a function on grid." class="img-fluid atto_image_button_text-bottom" height="285" src="https://k12.openstax.org/contents/raise/resources/a0f79653ec7d1fa67449d4d8872e608bcbc6e9a2" width="300"></p>
<ol class="os-raise-noindent">
  <li>
    What is the statement \(f(9) \approx 6\) saying about this situation?<br>
    <br>
    <strong>Answer:</strong> 9 years after 1977 (or in 1986), the cost in dollars for 1 watt of solar energy was about $6.

    <br>
    <br>
  </li>
  <li>
    What is \(f(4)\)? What about \(f(3.5)\)? What do these values represent in this context?<br>
    <br>
    <strong>Answer:</strong> \(f(4) \approx 25\), and \(f(3.5) \approx 30\). These values represent the cost in dollars of 1 watt of solar energy in 1981 (4 years after 1977) and halfway through 1980 (3.5 years after 1977).

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    <br>
  </li>
  <li>
    When \(f(t)=45\), what is \(t\)? What does that value of \(t\)&nbsp;represent in this context?<br>
    <br>
    <strong>Answer:</strong> 2 years after 1977 (in 1979), the cost in dollars of 1 watt of solar energy was $45.

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    <br>
  </li>
  <li>
    By what factor did the cost of 1 watt of solar energy change each year? (If you get stuck, consider creating a table.)<br>
    <br>
    <strong>Answer: </strong>\(\frac34\) or 0.75

  </li>
</ol>
<h4>Video: Understanding Characteristics of Graphs</h4>
<p>Watch the following video to learn more about the characteristics of graphs:&nbsp;</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
  <div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
      <source src="https://k12.openstax.org/contents/raise/resources/fd1b442820a507ecca84f874647c5a0400e67a8e">
      <track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/f7798ff0bec0471a3774432cfe2076f02a1f6b0c" srclang="en_us">https://k12.openstax.org/contents/raise/resources/fd1b442820a507ecca84f874647c5a0400e67a8e
    </video></div>
</div>
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<p>An imaginary line that a function value will not touch or cross is called an asymptote.</p>
<p>The value of the range that the function will never touch is called the horizontal asymptote.&nbsp;</p>
<p>In the function graphed, the horizontal asymptote is \(y=0\).</p>
<h4>Anticipated Misconceptions</h4>
<ul>
  <li>
    If students struggle with the function notation in the questions, ask them to recall what each part of \(f(t)\) means, or remind them that the \(f\) is the name of the function, and the \(t\)&nbsp;is the input value.
  </li>
</ul>
<h4>Activity Synthesis</h4>
<p>Focus the discussion on how students selected points on the graph to calculate the growth factor. Strategically, it is important to choose values of time \(t\) so that the price per watt \(f(t)\), in dollars, is unambiguous (for example, \(t = 0\), \(t=1\), and \(t=2\)). Because we are told that the price decreases exponentially, two values are enough to find the growth factor.</p>
<p>Interpreting statements such as \(f(4)=25\) in terms of the context is a chance to highlight precision in language, for example, to articulate that the 4 means 4 years after 1977 (1981), and the value \(f(4)=45\) means that each watt produced via solar energy cost $45 in 1981.</p>
<h3>5.9.2: Self Check</h3>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their understanding of the concepts explored in the activity</em>.</p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>The graph below shows the total smartphones, \(s\), in millions that were sold in the years, \(t\), since 2008. What does the point \((4, 125)\) represent in this context?</p>
<img alt="A graph. Total smartphone sales in millions. Years since 2008." src="https://k12.openstax.org/contents/raise/resources/a1949ca4d1812e197f8697dabef0793d67d63368"><br>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">Answers</th>
      <th scope="col">Feedback</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        In 2004, there were 125 million smartphones sold.
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: When \(t=4\), this means 4 years after 2008. The answer is in 2012, there were 125 million smartphones sold.
      </td>
    </tr>
    <tr>
      <td>
        In 2012, there were 125 million more smartphones than in 2008.
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: The \(s\) value represents the total number sold, not a comparison. The answer is in 2012, there were 125 million smartphones sold.
      </td>
    </tr>
    <tr>
      <td>
        In 2012, there were 125 million smartphones sold.
      </td>
      <td>
        That&rsquo;s correct! Check yourself: Since \(t=4\), the point represents 2012, 4 years after 2008. The \(s\) value is 125 and represents the number of millions of smartphones sold.
      </td>
    </tr>
    <tr>
      <td>
        In 2004, there were 125 million more smartphones than in 2000.
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: When \(t=4\), this means 4 years after 2008. The answer is in 2012, there were 125 million smartphones sold.
      </td>
    </tr>
  </tbody>
</table>
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<h3>5.9.2: Additional Resources</h3>
<p class="os-raise-text-bold"><em>The following content is available to students who would like more support based on their experience with the self check. Students will not automatically have access to this content, so you may wish to share it with those who could benefit from it.&nbsp;</em></p>
<h4>Reading Exponential Graphs</h4>
<p>The graph below describes the amount of caffeine, \(c\), in a person&rsquo;s body \(t\)&nbsp;hours after an initial measurement of 100 mg.</p>
<p><img alt="Graph of function." src="https://k12.openstax.org/contents/raise/resources/c6505a870a48b5a86ecc8bf01f3b25c1613990ba"></p>
<p>What does the point \(c(10)=35\) mean?</p>
<p>After 10 hours, there is 35 mg of caffeine in a person&rsquo;s body.</p>
<p>What is the value of \(t\) when \(c(t)=80\)?</p>
<p>When the caffeine concentration is 80 mg, time is 2 hours, so \(t=2\).</p>
<h4>Try It: Reading Exponential Graphs</h4>
<p>Using the graph above, what is the value of \(c(5)\), and what does it mean in the context of the situation?</p>
<p>Write down your answers. Then select the<strong> Solution </strong>button to compare your work.</p>
<h5> Solution</h5>
<p>Here is how to interpret this exponential graph:</p>
<p>When \(t=5\), the curve crosses at 60, so \(c(5)=60\).</p>
<p>This means that after 5 hours, the person has 60 mg of caffeine in their body.</p>