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<h4>Activity (20 minutes)</h4>
<p>This activity prompts students to look at how the \(y\)-intercept in slope-intercept form affects the graph of a
linear function. Circulate to ensure students are graphing their lines properly. If necessary, partner students during
the graphing process.</p>
<p>In pairs, students should discuss the vertical shifts to the parent function, \(f(x)=x\), to graph the new functions.
</p>
<h4> Launch</h4>
<p>Ask students if they have ever played a board game. Connect rolling dice and moving that number of spots to the
number of spots a line can shift up the graph.</p>
<p>For students who can benefit from a more accessible model, use geoboards and rubber bands. Use one color rubber band
for the \(x\)- and \(y\)-axes. Use a different color for each line graphed. Work with students to count spots on the
geoboard and discuss the graphs and their relationships.</p>
<h4>Student Activity </h4>
<h4>Activity</h4>
<h4>Vertical Shifts</h4>
<ol class="os-raise-noindent">
<li>Use the Desmos graphing tool or technology outside the course. Graph the following linear functions:</li>
</ol>
<ul>
<li>\(f(x)=x+4\)</li>
<li>\(f(x)=x+2\)</li>
<li>\(f(x)=x\)</li>
<li>\(f(x)=x-2\)</li>
<li>\(f(x)=x-4\)</li>
</ul>
<p>Compare your work:<br><img src="https://k12.openstax.org/contents/raise/resources/0120bf1497785e3d10342ceca2ddf50bf6a0d285" height="300"></p>
<ol class="os-raise-noindent" start="2">
<li>What are the slopes of each line?</li>
</ol>
<p>Correct! The slope of all the lines equals 1.</p>
<div class="os-raise-graybox">
<p>The parent function of a <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">linear function</span> is \(f(x)=x\).
<ul>
<li>The \(y\)-intercepts is (0,0).</li>
<li>The slope is 1.</li>
</ul>
</p>
</div>
<br>
<ol class="os-raise-noindent" start="3">
<li>How can the parent function, \(f(x)=x\) be shifted to overlap \(f(x)=x+2\)?</li>
</ol>
<p>Shift up 2.</p>
<ol class="os-raise-noindent" start="4">
<li>How can the parent function, \(f(x)=x\), be shifted to overlap \(f(x)=x-4\)?</li>
</ol>
<p>Shift down 4.</p>
<p>In \(f(x)=mx+b\), the \(b\) acts as the vertical shift. This is a type of transformation to the graph of \(f(x)=x\). </p>
<div class="os-raise-graybox">
<h5>Vertical Shift of a Function</h5>
<p>A vertical shift “transforms” the parent function into another function by moving the graph up or down \(d\) units. </p>
<p align="center">Vertical Shift → \(f(x) + d\)</p>
</div>
<br>
<p>If the \(d\) value is positive, the graph of the function shifts up. If the \(d\) value is negative, the graph of the function shifts down. </p>
<h4>Activity Synthesis</h4>
<p>Invite students to create other questions for students to answer about how much one linear function must shift to get
to another.</p>
<h3>4.11.2: 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>What transformation takes place from the graph of \(y=4x-3\) to \(y=4x +5\)?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>Down 8</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: For the second function to move down, there would be a
negative change. The answer is up 8.</p>
</td>
</tr>
<tr>
<td>
<p>Right 8</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: In \(y=mx+b\), the \(b\) describes the vertical shift. The
answer is up 8.</p>
</td>
</tr>
<tr>
<td>
<p>Up 8</p>
</td>
<td>
<p>That’s correct! Check yourself: In the two equations, the \(b\), or \(y\)-intercept of \(y=mx+b\),
changes. This determines the vertical change. From –3 to 5 is a vertical shift up 8.</p>
</td>
</tr>
<tr>
<td>
<p>Left 8</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: The \(y\)-intercept changing signifies a vertical change.
The answer is up 8.</p>
</td>
</tr>
</tbody>
</table>
<br>
<h4>4.11.2: 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>
<h4>Vertical Shifts and \(y\)-Intercepts</h4>
<p>In the equation \(f(x)=mx+b\):</p>
<ul>
<li>\(b\) is the \(y\)-intercept of the graph and indicates the point \((0,b)\) at which the graph crosses the
\(y\)-axis. </li>
</ul>
<ul>
<li>\(m\) is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run)
between each successive pair of points. </li>
</ul>
<p><img
alt="This graph shows how to calculate the rise over run for the slope on an x, y coordinate plane. The \(x\)-axis runs from negative 2 to 7. The \(y\)-axis runs from negative 2 to 5. The line extends right and upward from point (0,1), which is the \(y\)-intercepts. A dotted line extends two units to the right from point (0, 1) and is labeled Run = 2. The same dotted line extends upwards one unit and is labeled Rise =1."
src="https://k12.openstax.org/contents/raise/resources/fa8d440278a4d73105cd08756daceb00bac43939"></p>
<p>The \(y\)-intercept tells where the parent function, \(f(x)=x\), has shifted up or down.</p>
<p><strong>Example 1</strong></p>
<p>Tell how \(f(x)=x+7\) has shifted from the parent function \(f(x)=x\).<br>
<strong>Solution:</strong> Since \(b=7\), the \(y\)-intercept is +7, so the graph shifts up 7 from the parent function.</p>
<p> <strong>Example 2</strong></p>
<p>Tell how \(f(x)=x-9\) has shifted from the parent function \(f(x)=x\).<br>
<strong>Solution:</strong> Since \(b=-9\), the \(y\)-intercept is –9, so the graph shifts down 9 from the parent function.</p>
<p><strong>Example 3</strong></p>
<p>
Tell how \(f(x) = 2x + 1\) has shifted from \(f(x) = 2x + 5\).<br>
<strong> Solution:</strong> Since \(b = 1\), the \(y\)-intercept is 1. This point is 4 vertical units below the \(y\)-intercepts of the original line that had a \(y\)-intercept of 5. So, the graph shifts down 4 units from \(f(x) = 2x + 5\) to arrive at \(f(x) = 2x +1\).
</p>
<br>
<div class="os-raise-graybox">
<h5>Vertical Shift of a Function</h5>
<p>A vertical shift “transforms” the parent function into another function by moving the graph up or down \(d\) units. </p>
<p align="center">Vertical Shift → \(f(x) + d\)</p>
</div>
<br>
<p>If the \(d\) value is positive, the graph of the function shifts up. If the \(d\) value is negative, the graph of the function shifts down. </p>
<h4>Try It: Vertical Shifts and \(y\)-Intercepts</h4>
<p>Tell the shift made to \(f(x)=x\) to arrive at \(f(x)=x - 4\).</p>
<p>Write down your answer. Then select the <strong>solution </strong>button to compare your work.<br>
</p>
<p><strong>Answer:</strong></p>
<p>Here is how to find the shift made to \(f(x)=x\):</p>
<p>Since the \(b\) of slope-intercept form is –4, this is also the \(y\)-intercept. Therefore, the graph of
\(f(x)=x\) is shifted down 4 to arrive at \(f(x)=x-4\).</p>