c38f248c-675d-4b19-a888-075807107956.html

<h4>Activity (15 minutes)</h4>
<p>This activity reinforces the understandings that students began to develop in an earlier lesson about the
  connections between the structure of two-variable linear equations, their graphs, and the situations they represent. </p>
<p>Students first practice relating the parameters of an equation in slope-intercept form to the features of the graph
  and interpreting them in terms of the situation (TEKS A.1(B)). Next, they practice making a case for how they know that a
  graph represents an equation given in standard form.</p>
<p>Some students may argue that substituting the \( (x,y) \) pair of any point
  on the line gives a true statement, suggesting that the graph does match the equation. Or they may reason about the
  points on the graph in terms of almonds and figs and come to the same conclusion. For example, \( (8,3) \) and \(
  (11,1) \) are points
  on the line. If Clare buys 8 pounds of almonds and 3 pounds of figs, or 11 pounds of almonds and 1 pound of figs,
  the price is $75.</p>
<p>Ask these students how they would check whether the points with fractional \( x\) - and \( y\) -values (which are
  harder to identify precisely from
  the graph) would also produce true statements when those values are substituted. Use this difficulty to motivate
  rearranging the equation into slope-intercept form.</p>
<p>The work in this activity requires students to reason quantitatively and abstractly about the equation and the
  graph and to construct a logical argument.</p>
<h4>Launch</h4>
<p>Display the two graphs in the task statement for all to see. Tell students the graphs represent two situations they
  have seen in earlier activities.</p>
<p>Arrange students in groups of 2. Give students a minute of quiet time to think individually and ask them to be
  prepared to share at least one thing they notice and one thing they wonder about the graphs. Give them another
  minute to discuss their observations and questions with their partner before moving on to the task.</p>
<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 8 Discussion Supports: Speaking</p>
  </div>
  <div class="os-raise-extrasupport-body">
      <p>Support students in producing statements when they share
    their observations. Provide sentence frames for students to use when they are sharing, such as &ldquo;I noticed
    that . . .&rdquo; or &ldquo;I wondered about _____ because . . .&rdquo;</p>
      <p class="os-raise-text-italicize">Design Principle(s): Support
    sense-making</p>
</div></div>
<br>
<h4>Student Activity</h4>
<p>For this activity, you will use two graphs that represent situations you saw in other activities. </p>
<p>For questions 1–3, use the graph below:<br>
  The graph represents \(a = 450 − 20t\), which describes the relationship between gallons of water in a tank and time
  in minutes.</p>
<img alt="Graph of a line. Vertical axis, amount in tank, gallons. Horizontal axis, time, minutes." height="195"
  src="https://k12.openstax.org/contents/raise/resources/627ff6c21d4900907647e8abc921dfb1111f374a" width="244"><br>
<br>
<ol class="os-raise-noindent" >
  <li>Where on the graph can we see the 450?</li>
</ol>
<p><strong>Answer:</strong><br>
  450 is where the graph intersects the
  vertical axis.</p>
<ol class="os-raise-noindent"  start="2">
  <li>Where can we see the -20? </li>
</ol>
<p><strong>Answer:</strong><br>
  −20 is the slope.</p>
<ol class="os-raise-noindent"  start="3">
  <li>What do the numbers 450 and -20 mean in the situation that is graphed?</li>
</ol>
<p><strong>Answer:</strong><br>
  Your answer
  may vary, but here is a sample. <br>
  450 is the gallons of water in the tank at 0 minutes, before the tank starts to be drained. −20
  means the gallons of water (vertical value) drops by 20 for every 1 minute increase in time (horizontal value). </p>
<ol class="os-raise-noindent"  start="4">
  <li> The graph represents \(6x +9y = 75\). It describes the relationship between pounds of almonds and figs and the
    dollar amount Clare spent on them. Suppose a classmate says, “I am not sure the graph represents \(6x +9y = 75\)
    because I don&rsquo;t  see the 6, 9, or 75 on the graph.” How would you show your classmate that the graph indeed
    represents
    this equation? <br>
    <br>
    <img alt="Graph of a line. Vertical axis, dried figs, pounds. Horizontal axis, Almonds, pounds." height="196"
      src="https://k12.openstax.org/contents/raise/resources/e183cfb486b4c8793f50d9d287a0c65abc863bad"
      width="239"> </li>
</ol>
<p><strong>Answer:</strong><br>
  Your answer may vary, but here is a sample.</p>
<ul>
  <li> If we substitute 0 for \(x\) in the equation and solve for \(y\), we get \(\frac {75}{9}\) or \(8 \frac 13\) . That combination is the point
    \((0,8 \frac 13)\) or the \(y\)-intercept of the graph. If we substitute 0 for \(y\) in the equation and solve for \(x\), we
    get 12.5 for \(x\). That combination is the point (12.5,0) or the \(x\)-intercept of the graph.</li>
  <li>If we substitute the \(x\)- and \(y\)-values of any point on the graph in the equation, the equation remains true.</li>
  <li>If we rewrite the equation and solve for \(y\), we have \( y = \dfrac {75 - 6x}{9} \) or \( y = 8\frac13 - \frac23x \). The \( 8\frac13 \) matches where the graph
    intersects the \(y\)-axis and \( -\frac23\) matches the slope of the
    graph. (For every 3 additional pounds of almonds that Clare bought, she could buy 2 fewer pounds of figs.)</li>
  <li>From the graph, we can see that the \(y\)-intercept is \( (0,8\frac13) \) and the slope is \( -\frac23 \), so the equation of the line is
    \( y = 8\frac13 - \frac23x \). Multiplying each side of the equation by 9 (an acceptable move) gives an equivalent equation, \( 9y = 75 - 6x\), 
    which can be rewritten as \( 6x + 9y = 75\).</li>
</ul>
<h4>Activity Synthesis</h4>
<p> Focus the discussion on students&rsquo; explanations for the last question. If no one mentions that
  \( 6x+9y=75 \)   can be rearranged into an equivalent equation,
  \( y=8\frac13 -\frac23 x \)  , point this out. (Demonstrate the
  rearrangement process, if needed.)</p>
<p>Ask students if we can now see the  \( 8\frac13 \)  and the \( - \frac23 \)  on the graph and if so, where they are visible. To help students
  connect these values back to the quantities in the situation, ask what each value tells us about almonds and figs.
  Make sure students see that the  \( 8 \frac13 \)  tells us that if Clare
  bought no almonds, she could buy  \( 8\frac13 \)  pounds of figs. For every
  pound of almonds she buys, she can buy less figs&mdash; \( \frac23 \)  pound
  less, to be exact.</p>
<h4>1.11.2 Self-Check</h4>
<p class="os-raise-text-bold"><em>Following 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>For the line graphed below, what is the
  \(y\)-intercepts?</p>
<img alt class="atto_image_button_text-bottom" height="365" role="presentation"
              src="https://k12.openstax.org/contents/raise/resources/513c368565be1a5e7eac0e7aaec73f2eecc2d47b"
              width="325"><br>
<br>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">Answers</th>
      <th scope="col">Feedback</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>\( (0, -5) \)</td>
      <td>That&rsquo;s correct! Check yourself: The \(y\)-intercepts is where the graph crosses the vertical, when \(
        x=0 \).</td>
    </tr>
    <tr>
      <td>\( (-5,0) \)</td>
      <td>Incorrect. Let&rsquo;s try again a different way: This is the \( x \)-intercept. Look at where the line
        crosses the vertical axis. The answer is \( (0, -5) \).</td>
    </tr>
    <tr>
      <td>\((0, 5)\)</td>
      <td>Incorrect. Let&rsquo;s try again a different way: Look at where the line crosses the vertical axis. It
        crosses below the \( x \)-axis. The sign should be negative. The answer is \( (0, -5) \).</td>
    </tr>
    <tr>
      <td>\( (5, 0) \)</td>
      <td>Incorrect. Let&rsquo;s try again a different way: Look at where the line crosses the vertical axis. It
        crosses below the \( x \)-axis. Also, a coordinate is written \( (x, y) \). The answer is \( (0, -5) \). </td>
    </tr>
  </tbody>
</table>
<br>
<h4>1.11.2: Additional Resources</h4>
<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>
<br>
<h4>Finding Intercepts from Graphs and Equations</h4>
<p>The points where a line crosses the \( x \)-axis and the \( y \)-axis are called the
  intercepts of the line.</p>
<p> Let&rsquo;s look at the graphs of the lines. </p>
<img
      alt="The figure shows four graphs of different equations. In example a the graph of 2 x plus y plus 6 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, 6) and (3, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example b the graph of 3 x minus 4 y plus 12 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, negative 3) and (4, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example c the graph of x minus y plus 5 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, negative 5) and (5, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example d the graph of y plus negative 2 x is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The point (0, 0) is plotted and labeled. A straight line goes through this point and the points (negative 1, 2) and (1, negative 2) and has arrows on both ends."
      class="img-fluid atto_image_button_text-bottom" height="612"
      src="https://k12.openstax.org/contents/raise/resources/fe7c894af22fc816272cc2a3a72e056a1da78b83" width="550"><br>
<br>
<p>First, notice where each of these lines crosses the \( x\) -axis.</p>
<p>Now, let&rsquo;s look at the points where these lines cross the \( y \)-axis.</p>
<table class="os-raise-wideequaltable">
  <thead>
    <tr>
      <th scope="col">Figure</th>
      <th scope="col">The Line Crosses the \(x\)-axis at:</th>
      <th scope="col">Ordered Pair for this Point</th>
      <th scope="col">The Line Crosses the \(y\)-axis at:</th>
      <th scope="col">Ordered Pair for this Point</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Figure (a)</td>
      <td>3</td>
      <td>\((3,0)\)</td>
      <td>6</td>
      <td>\((0,6)\)</td>
    </tr>
    <tr>
      <td>Figure (b)</td>
      <td>4</td>
      <td>\((4,0)\)</td>
      <td>-3</td>
      <td>\((0,-3)\)</td>
    </tr>
    <tr>
      <td>Figure (c)</td>
      <td>5</td>
      <td>\((5,0)\)</td>
      <td>-5</td>
      <td>\((0,5)\)</td>
    </tr>
    <tr>
      <td>Figure (d)</td>
      <td>0</td>
      <td>\((0,0)\)</td>
      <td>0</td>
      <td>\((0,0)\)</td>
    </tr>
    <tr>
      <td>General Figure</td>
      <td>\(a\)</td>
      <td>(\(a,0)\)</td>
      <td>\(b\)</td>
      <td>(0,\(b)\)</td>
    </tr>
  </tbody>
</table>
<br>
<p class="os-raise-text-bold">\(X\)-INTERCEPT and \(Y\)-INTERCEPT of a Line</p>
<p> The  \(x\) -intercept is the point  \( (a , 0) \) where the line crosses the  \(x\)-axis. <br>
  The
  \(y\)-intercept is the point \( (0 , b) \) where the line crosses the \(y\)-axis.</p>
<img
        alt="The table has 3 rows and 2 columns. The first row is a header row with the headers x and y. The second row contains a and 0. The third row contains 0 and b."
        class="img-fluid atto_image_button_text-bottom" height="92"
        src="https://k12.openstax.org/contents/raise/resources/cfa22a98c89816890def2a20a309f5e74ebe366d"
        width="429"><br>
<br>
<p class="os-raise-text-bold">Find the \( x \)- and \( y \)-intercepts from the Equation of a Line</p>
<p> Use the equation of the line. To find: </p>
<ul>
  <li> the \(x\)-intercept of the line, let \( y = 0 \) and solve for \( x \). </li>
  <li> the \(y\)-intercept of the line, let \( x = 0 \) and solve for \( y \). </li>
</ul>
<p class="os-raise-text-bold">Example</p>
<p>Find the intercepts of \( 2x + y = 8 \)</p>
<p class="os-raise-text-bold"><strong>Answer:</strong></p>
<p>To find the \(x\)-intercept:<br>
  <strong>Step 1</strong> - Let \(y = 0\).<br>
\(2x + 0 = 8\)</p>
<p><strong>Step 2 </strong>- Solve for \(x\).<br>
\(x = 4\)</p>
<p><strong>Step 3</strong> - Write the intercept as a point.<br>
\((4,0)\)</p>
<br />
<p>To find the \(y\)-intercept:<br>
  <strong>Step 1</strong> - Let \(x = 0\).<br>
\(2(0) + y = 8\)</p>
<p><strong>Step 2</strong> - Solve for \(y\).<br>
\(y=8\)</p>
<p><strong>Step 3 </strong>- Write the intercept as a point.<br>
\((0, 8)\)</p>
<br>

<h4>Try It: Finding Intercepts from Graphs and Equations</h4>
<p>Find the \(x\)- and \(y\)-intercept of \(x+4y=8\).</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer: </strong>(8, 0) (0, 2)</p>
<p>Here is how to find the intercepts using a general strategy.</p>
<p>To find the \( x \)-intercept, <br>
  <strong>Step 1</strong> Let \( y=0 \).<br>
\( x+4(0)=8\)</p>
<p><strong>Step 2</strong> - Solve for \( x \).<br>
\(x = 8\)</p>
<p><strong>Step 3</strong> -  Write the intercept as a point.<br>
\((8, 0)\)</p>
<br>

<p>To find the \(y\)-intercept:<br>
  <strong>Step 1 </strong>- let \( x=0 \)<br>
\((0)+ 4y = 8\)</p>
<p><strong>Step 2</strong> - Solve for \( y \).<br>
\(y = 2\) </p>
<p><strong>Step 3</strong> - Write the intercept as a point.<br>
\((0, 2)\)</p>