141698a9-1e1b-43e0-806f-3a375cd0ce1b.html

<h4>Activity (15 minutes)</h4>
<p>Earlier, students were given quadratic expressions of the form \(ax^2+ bx + c\) where the \(a\) is not 1. They found
  that the rewriting process was a bit more involved but was not impossible, at least for the problems at hand.</p>
<p>In this activity, they encounter a real-world example in which they struggle to find a combination of rational
  factors of \(a\) and \(c\) that would produce the value of \(b\). (Realistic quadratic functions don't always have
  rational numbers for their zeros, so the quadratic expressions that define them cannot always be written in factored
  form. At this point, students are not yet considering rational and irrational solutions and are not expected to know
  why some expressions cannot be easily written in factored form.)</p>
<p>By graphing the function, students discover that the horizontal intercepts are decimals (rounded to different decimal
  places, depending on the graphing technology used). The graph allows them to estimate the solution, but that is as far
  as they could go. The challenges in this activity set the stage for introducing a more productive technique for
  solving quadratic equations.</p>
<h4>Launch</h4>
<p>Keep students in groups of two. Let students become briefly frustrated by their unsuccessful attempts to find factors
  of the expression in standard form, but move them on to the last question after a few minutes. Provide access to
  devices that can run Desmos or other graphing technology.</p>
<br>
<div class="os-raise-extrasupport">
  <div class="os-raise-extrasupport-header">
    <p class="os-raise-extrasupport-title">Support for English Language Learners</p>
    <p class="os-raise-extrasupport-name">MLR 6 Three Reads: Reading </p>
  </div>
  <div class="os-raise-extrasupport-body">
    <p>Use this routine to support reading comprehension, without solving, for
      students. Use the first read to orient students to the situation. Ask students to describe what the situation is
      about without using numbers. (An engineer is designing a water fountain.) Use the second read to identify quantities
      and relationships. Ask students what can be counted or measured without focusing on the values. Listen for, and
      amplify, the important quantities that vary in relation to each other in this situation: height of a drop of water,
      in meters, and time, in seconds. After the third read, ask students to brainstorm possible strategies to answer the
      questions. This helps students connect the language in the word problem and the reasoning needed to solve the
      problem.</p>
    <p class="os-raise-text-italicize">Design Principle: Support sense-making</p>
    <p class="os-raise-extrasupport-title">Learn more about this routine</p>
    <p>
      <a href="https://www.youtube.com/watch?v=Q2PGJThrG2Q" target="_blank">View the instructional video</a>
      and
      <a href="https://k12.openstax.org/contents/raise/resources/bf750b41e6483d334d575e1d950851bfa07cfd26" target="_blank">follow along with the materials</a>
      to assist you with learning this routine.
    </p>
    <p class="os-raise-extrasupport-title">Provide support for students</p>
    <p>
      <a href="https://k12.openstax.org/contents/raise/resources/06eafa198e5345452a0adbc81731e7383785aa42" target="_blank">Distribute graphic organizers</a>
      to the students to assist them with participating in this routine.
    </p>
  </div>
</div>
<br>
<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>
  <div class="os-raise-extrasupport-body">
    <p>
      Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems, and other text-based content. Clarify any unfamiliar terms or phrases. </p>
    <p class="os-raise-text-italicize">Supports accessibility for: Language; Conceptual processing</p>
  </div>
</div>
<br>
<h4>Student Activity </h4>
<p>An engineer is designing a fountain that shoots out drops of water. The nozzle from which the water is launched is 3
  meters above the ground. It shoots out a drop of water at a vertical velocity of 9 meters per second.</p>
<p>Function \(h\) models the height in meters, \(h\), of a drop of water \(t\) seconds after it is shot out from the
  nozzle. The function is defined by the equation \(h(t) = -5t^2+ 9t + 3\).</p>
<p>How many seconds until the drop of water hits the ground?</p>
<ol class="os-raise-noindent">
  <li>Write an equation that we could solve to answer the question.</li>
</ol>
<p><strong>Answer:</strong> \(-5t^2+ 9t + 3 = 0\)</p>
<ol class="os-raise-noindent" start="2">
  <li>Try to solve the equation by writing the expression in factored form and using the zero product property. If it
    cannot be factored, explain how you came to this conclusion.</li>
</ol>
<p><strong>Answer:</strong> There is no combination of integer factors of the first coefficient, -5, and the third
  coefficient, 3, that would result in a middle term of 9. Because it is prime, the expression cannot be written in
  factored form.</p>
<ol class="os-raise-noindent" start="3">
  <li>Solve the equation by graphing the function.</li>
</ol>
<p>Use the Desmos graphing tool or technology outside the course.</p>
<p>Explain how you found the solution.</p>
<p><strong>Answer:</strong> About 2. The graph shows two horizontal intercepts, one with a positive \(x\)-coordinate and
  one negative \(x\)-coordinate. The negative one does not apply here because time in seconds cannot be a negative
  value. The other horizontal intercept is around \((2.087,0)\). This means it takes about 2 seconds for the drop to hit
  the ground.</p>
<p><img height="400" src="https://k12.openstax.org/contents/raise/resources/7cd52056b779788ec9c2b1b4f86cf4a673f8f602" width="400"></p>
<h4>Using Technology to Find the Rational Factors</h4>
<p>Watch the following video to learn more about using technology to find the rational factors of a quadratic equation.
</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/c4a36219b2c62fa8ffe57f47a507437fcb41b12f">
      <track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/f6efd506061a68e0b17047645681033370565600" srclang="en_us">
      https://k12.openstax.org/contents/raise/resources/c4a36219b2c62fa8ffe57f47a507437fcb41b12f
    </video></div>
</div>
<br>
<br>
<h4>Activity Synthesis</h4>
<p>Ask students to share some challenges they came across when trying to rewrite the expressions in factored form.
  Solicit some ideas about why this equation presented those challenges. Then, discuss how they found or estimated the
  solution by graphing.</p>
<p>The approximate solution to the equation, given by the zero of the function and the \(x\)-intercept of the graph, is
  2.087 seconds. Some graphing tools would give an approximation with a longer decimal expansion, giving a clue that it
  might be trickier to rewrite the equation in factored form. After all, when finding factors, we usually look for
  integers. (Some quadratic expressions containing non-integer rational numbers can still be written in factored form.
  For example, \(x^2 + \frac {3}{4} + \frac {1}{8}\) can be written as \((x + \frac {1}{2})(x + \frac {1}{4})\).)</p>
<p>Highlight that some equations are difficult or impossible to rewrite in factored form. In fact, when quadratic models
  appear in real life, this is usually the case. Graphing is a way to solve these equations, but there are other
  techniques, which students will learn over the next several lessons.</p>
<h3>8.10.3: 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>
<br>
<p class="os-raise-text-bold">QUESTION:</p>
<p>A tennis ball is hit by a racket.</p>
<p>The function \(h(t) = -3t^2+ 7t + 3\) models the height in meters, \(h\), of the tennis ball \(t\) seconds after the
  impact with the racket.</p>
<p>Use graphing technology to determine the amount of time that has passed from the tennis ball being hit to when it
  reaches the ground.</p>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">
        Answers
      </th>
      <th scope="col">
        Feedback
      </th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        <p>1.167 seconds</p>
      </td>
      <td>
        <p>Incorrect. Let's try again a different way: This answer is the time it takes for the tennis ball to reach its
          maximum height. The answer is 2.703 seconds when the height is back to 0.</p>
      </td>
    </tr>
    <tr>
      <td>
        <p>2.703 seconds</p>
      </td>
      <td>
        <p>That's correct! Check yourself: Substitute 2.703 for \(t\) and check that the function is approximately 0.
          \(-3(2.703)2 + 7(2.703) + 3 &asymp; 0\). The answer is correct.</p>
      </td>
    </tr>
    <tr>
      <td>
        <p>0.37 seconds</p>
      </td>
      <td>
        <p>Incorrect. Let's try again a different way: The value -0.37 is a zero for the function, but a negative time
          value is not relevant in this example. The answer is 2.703 seconds when the height is back to 0 at a positive
          \(t\).</p>
      </td>
    </tr>
    <tr>
      <td>
        <p>3.124 seconds</p>
      </td>
      <td>
        <p>Incorrect. Let's try again a different way: Check to see if you entered the function correctly into the
          graph. The answer is 2.703 seconds when the height is back to 0 at a positive \(t\).</p>
      </td>
    </tr>
  </tbody>
</table>
<br>
<h3>8.10.3: 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.</em></p>
<h4>Using Technology to Find Rational Factors</h4>
<p>Let's use graphing technology to solve a situation involving a quadratic function that cannot be solved using
  factoring.</p>
<p>A thick steel support cable at a construction site forms an arch-like shape. It starts attached to the top of a
  column, goes up through a loop at a high point, and then back down to the ground.</p>
<p><img height="391" src="https://k12.openstax.org/contents/raise/resources/653cb5cbeb27d9ebd6051259d7127fa2ad63c627" width="400"></p>
<p>The height of the cable in meters is represented by the function \(h(x) = -4x^2+ 16x + 5\) where \(x\) is the
  horizontal distance from the column, in meters.</p>
<p>How far away from the column does the cable attach to the ground?</p>
<p>The first step in solving would be to set the function equal to 0 and try to factor. Since this equation cannot be
  factored, we will use graphing technology.</p>
<p>Let's graph the function using the equation \(y = -4x^2+ 16x + 5\).</p>
<p><img alt="GRAPH OF A DOWNWARD PARABOLA WITH A POSITIVE \(x\)-intercepts AT 4.291." height="413" src="https://k12.openstax.org/contents/raise/resources/56a403c649b8b302851a397b794f3d103c41113c" width="400"></p>
<p>We can see the function has a zero at \(x=4.291\). The \(x\)-intercept is \((4.291, 0)\).</p>
<p>This means the cable attaches to the ground 4.291 meters away from the column.</p>
<p>What else about the steel cable system do we know from this graph?</p>
<p>We can also tell that the cable attaches to the column 5 meters off the ground at the \(y\)-intercept and the loop is
  approximately 21 meters high at the vertex of the parabola.</p>
<h4>Try It: Using Technology to Find Rational Factors</h4>
<p>The height, in meters, of a model rocket being launched is modeled by the function \(h(t) = -2t^2 + 14t + 2\) where
  \(t\) is the number of seconds after launch.</p>
<p>How many seconds pass before the rocket reaches the ground?</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to solve this problem using graphing technology:</p>
<p>Since the expression cannot be factored, it should be graphed.</p>
<p><img alt="GRAPH OF A DOWNWARD PARABOLA WITH A POSITIVE \(x\)-intercepts AT 7.14." height="405" src="https://k12.openstax.org/contents/raise/resources/98fe8e4e5f929935e52a0612e27e3aed2aecf0ac" width="400"></p>
<p>The function has a zero at \(x=7.14\). The \(x\)-intercept is \((7.14, 0)\).</p>
<p>This means the rocket travels for 7.14 seconds before reaching the ground.</p>