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<h4>Activity (20 minutes)</h4>
<p>This activity is an extension of the expectations in the TEKS.</p>
<p>In this activity, students learn to use graphing technology to graph an equation in function notation, evaluate the
function at a specific input value, create a table of inputs and outputs, and identify the coordinates of the points
along a graph. They also revisit how to set an appropriate graphing window.<br></p>
<h4>Launch</h4>
<p>Display the equation \(B(x)=10x+25\) for all to see. Ask students:</p>
<ul>
<li>“How would you go about finding the value of \(B(1.482)\)?” (Substitute 1.482 into the expression
\(10x+25\) and evaluate, or use the graph of \(y=10x+25\) to estimate the \(y\)-value when
\(x\) is 1.482.)</li>
<li>“How would you find the value of \(x\) that makes \(B(x)=103.75\) true? Try solving the equation."
(Solve \(10x+25=103.75\), or use the graph of \(y=10x+25\) to estimate the \(x\)-value when \(y\) is
103.75.)</li>
</ul>
<p>Explain that while it’s possible to use a graph to find or estimate unknown input or output values, it is hard
to be precise when using a hand-drawn or printed graph. We can evaluate the expression or solve the equation
algebraically, but computing by hand can get cumbersome (though a calculator can take care of the most laborious
part). Let's see how graphing technology can help us!</p>
<p>The digital version of this activity includes instructions for using the Desmos Math Tool to graph an equation,
change the graphing window, and find the output of a function for a given input. If students will be using different
graphing technology, consider preparing alternate instructions.</p>
<ul>
<li>Demonstrate how to graph \(B(x)=10x+25\), change the graphing window, and find \(B(4)\) on the
graph.</li>
<li>Ask students to find the value of \(x\) when \(B(x)\) is 100. Then, demonstrate some ways (other
than approximating visually) to solve for \(x\) given \(B(x)=100\):
<ol class="os-raise-noindent">
<li>Trace the line. Coordinates appear as we move along the line. Stop when the \(y\)-coordinate is 100 and see
what the \(x\)-coordinate is.</li>
<li>Type \(y=100\) in the expression list. A horizontal line appears. Find the intersection of this line and
the graph of \(B\).</li>
</ol>
</li>
</ul>
<p>Ask students to use one or both of these strategies to complete the activity.</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">Action and Expression: Provide Access for Physical Action</p>
</div>
<div class="os-raise-extrasupport-body">
<p> Support effective and efficient use of tools
and assistive technologies. To use graphing technology, some students may benefit from a demonstration or access to
step-by-step instructions.</p>
<p class="os-raise-text-italicize">Supports accessibility for: Organization; Memory; Attention</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<br>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div><br>
<p>The function \(B\) is defined by the equation \(B(x)=10x+25\). Use the graphing tool here or technology outside the course. Use your graph to find the value of each expression below. </p>
<p>If needed, remember -</p>
<ul>
<li>You can collapse the equation entry tab by selecting the button.</li>
<li>To change the scale along either the x- or \(y\)-axis, click on the wrench button and then enter minimum and maximum values for each axis and set the step value.</li>
<li>To find a point on the line, click on the line and drag the mouse along the line to display the ordered pairs for each location.</li>
</ul>
<ol class="os-raise-noindent" type="1">
<li>What is the value of \(B(6)\)?</li>
<p><strong>Answer:</strong> \(B(6) = 85\)</p>
<li>What were the window values you used for the \(x\)-axis? (Write the values as an inequality.)</li>
<p><strong>Answer:</strong> \(-1 \leq x \leq 7\)</p>
<li>What were the window values you used for the \(y\)-axis? (Write the values as an inequality.)</li>
<p><strong>Answer:</strong> \(-1 \leq y \leq 100\)</p>
<li>What is the value of \(B(2.75)\)?</li>
<p><strong>Answer:</strong> \(B(2.75) = 52.5\)</p>
<li>What is the value of \(B(1.482)\)?</li>
<p><strong>Answer:</strong> \(B(1.482) = 39.8\)</p>
<li>What is the value of \(x\) if \(B(x) = 93\)?</li>
<p><strong>Answer:</strong> \(x = 6.8\)</p>
<li>What were the window values you used for the \(x\)-axis? (Write the values as an inequality.)</li>
<p><strong>Answer:</strong> \(-2 \leq x \leq 16\)</p>
<li>What were the window values you used for the \(y\)-axis? (Write the values as an inequality.)</li>
<p><strong>Answer:</strong> \(-15 \leq y \leq 150\)</p>
<li>What is the value of \(x\) if \(B(x) = 42.1\)?</li>
<p><strong>Answer:</strong> \(x = 1.71\)</p>
<li>What is the value of \(x\) if \(B(x) = 116.25\)?</li>
<p><strong>Answer:</strong> \(x = 9.125\)</p>
</ol>
<h4>Activity Synthesis</h4>
<p>Invite students to share any insights they had while using the graphing tool and techniques to evaluate expressions
and solve equations. In what ways might the tool and techniques be handy? When might they be limited?</p>
<p>Also discuss any issues that students encountered while completing the task—technical or otherwise.</p>
<p>If desired, consider showing another way to obtain input-output pairs of a function in Desmos.</p>
<p>Let’s assign a new input variable, say \(n\), to function \(B\). If we enter \(n,B(n)\) in the expression
list, activate a slider for \(n\), and enable the option to label points, the graph will show the coordinate pair
for any value of \(n\).</p>
<p><img alt="Screenshot of a Desmos graph."
src="https://k12.openstax.org/contents/raise/resources/68a0d2620d0158fbeff76e143ba9bf8628599d41" width="400"></p>
<div class="os-raise-graybox">
<h4>Support for English Language Learners</h4>
<p>Give students additional time to share their insights in small groups before the whole-class discussion. Invite
groups to discuss the advantages and disadvantages of using the graphing tool and to rehearse what they will say
when they share with the whole class. Rehearsing provides students with additional opportunities to speak and
clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
</p>
</div><br>
<h3>4.5.4: 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 function \(D\) is defined by the equation \(D(x)=2.5x+12\).</p>
<p dir="ltr">Which describes the window that can be used to find the value of \(x\) when \(D(x)=23.25\) using
graphing technology and gives the correct value of \(x\)?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
Graph \(y=2.5x+12\), \(x=23.25\); window: \(x\) from 0 to 30, \(y\) from 0 to 100; \(x=70.125\)
</td>
<td>
Incorrect. Let’s try again a different way: The value 23.25 is the output value, not the input value.
The window must include the value of \(D(x)\) and the estimated value of \(x\). The answer is: Graph
\(y=2.5x+12\), \(y=23.25\); window: \(x\) from 0 to 10, \(y\) from 20 to 25; \(x=4.5\).
</td>
</tr>
<tr>
<td>
Graph \(y=2.5x+12\), \(y=23.25\); window: \(x\) from 0 to 10, \(y\) from 20 to 25; \(x=4.5\)
</td>
<td>
That’s correct! Check yourself: Graph each equation using a window for \(y\) close to the given value
and a window for \(x\) close to an estimated value. The value of \(x\) where the lines meet is 4.5.
</td>
</tr>
<tr>
<td>
Graph \(y=2.5x\), \(y=23.25\); window \(x\) from 0 to 10, \(y\) from 20 to 25; \(x=9.3\)
</td>
<td>
Incorrect. Let’s try again a different way: be sure to graph the full equation for \(D(x)\), using a
window for \(y\) close to the given value and a window for \(x\) close to an estimated value. The answer is:
Graph \(y=2.5x+12\), \(y=23.25\); window: \(x\) from 0 to 10, \(y\) from 20 to 25; \(x=4.5\).
</td>
</tr>
<tr>
<td>
<p>Graph \(y=2.5x+12\), \(y=23.25\); window: \(x\) from 20 to 25, \(y\) from 0 to 10; \(x=4.5\)
</td>
<td>
Incorrect. Let’s try again a different way: Make sure to use a window where you can see that given
value of \(y\) and the estimated value of \(x\). The answer is: Graph \(y=2.5x+12\), \(y=23.25\);
window: \(x\) from 0 to 10, \(y\) from 20 to 25; \(x=4.5\).
</td>
</tr>
</tbody>
</table>
<br>
<h3>4.5.4: 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>Finding Missing Values</h4>
<p>Function \(G\) is defined by \(G(x)=5x+20\). Find the following values using graphing technology.</p>
<ol class="os-raise-noindent" start="1" type="a">
<li>\(G(2)\)</li>
</ol>
<p>To find \(G(2)\), look at \(x=2\) on the \(x\)-axis on the graph of \(G(x)\). Use proper steps with your graphing
technology to find the \(y\)-value on the line: \(G(2)=30\).<br>
<img alt="GRAPH OF A STRAIGHT LINE WITH \(y\)-intercepts OF 20 THAT PASSES THROUGH THE POINT (2, 30)."
class="img-fluid atto_image_button_text-bottom" height="307"
src="https://k12.openstax.org/contents/raise/resources/53b0a95f8355556121169eeec75706012c7a0b9e" width="300">
</p>
<ol class="os-raise-noindent" start="2" type="a">
<li>\(G(x)=32\)</li>
</ol>
<p>To find the value of \(x\) where \(G(x)=32\), look at where the graph crosses the \(y\)-value of 32. Use proper steps
with your graphing technology to find the \(x\)-value where the graph meets \(y=32\).</p>
<p><img
alt="GRAPH OF A STRAIGHT LINE WITH \(y\)-intercepts OF 20 THAT PASSES THROUGH THE POINT (2.4, 32). A GREEN HORIZONTAL LINE THAT PASSES THROUGH (0, 32)."
class="img-fluid atto_image_button_text-bottom" height="307"
src="https://k12.openstax.org/contents/raise/resources/0b8db35d10d091627f644e451f3267ab28421acc" width="300"></p>
<p>\(G(2.4)=32\)</p>
<h4>Try It: Finding Missing Values</h4>
<p>Function \(H\) is defined by \(H(x)=2x+4\). Find \(H(x)=10.4\) using graphing technology.</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 find the value of \(H(x)\) using graphing technology:<br>To find the value of \(x\)
where \(H(x)=10.4\), look at where the graph of \(H(x)=2x+4\) crosses the \(y\)-value of 10.4. Use
proper steps with your graphing technology to find the \(x\)-value where the graph meets \(y=10.4\).<br></p>
<p>\(H(3.2)=10.4\)</p>
<p><img
alt="GRAPH OF A STRAIGHT LINE WITH \(y\)-intercepts OF 4 THAT PASSES THROUGH THE POINT (3.2, 10.4). A RED HORIZONTAL LINE THAT PASSES THROUGH (0, 10.4)"
class="img-fluid atto_image_button_text-bottom" height="309"
src="https://k12.openstax.org/contents/raise/resources/84083ff5ffe53389d428f550de6048a04cf9f7dd" width="300">
</p>