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<h4>Activity (25 minutes)</h4>
<p>A table for the relation \(f\) is below. </p>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
<tr>
<th scope="row">\(x\)</th>
<td>
1
</td>
<td>
2
</td>
<td>
3
</td>
<td>
1
</td>
<td>
4
</td>
<td>
5
</td>
</tr>
<tr>
<th scope="row">\(f(x)\)</th>
<td>
-1
</td>
<td>
5
</td>
<td>
-2
</td>
<td>
1
</td>
<td>
3
</td>
<td>
-2
</td>
</tr>
</tbody>
</table>
<br>
<ol class="os-raise-noindent" start="1" type="1">
<li>Is this relation a function? Explain your answer.</li>
<p> <strong>Answer: </strong> \(f\) is not a function because \(f(1) =-1\) and \(f(1)=1\).</p>
<li>Interact with the graph below. Graph the points in the relation on a coordinate grid. </li>
<p> <strong>Answer: </strong> <img alt="COORDINATE GRID FROM −1 TO 8 ON THE X-AXIS AND FROM −2 TO 6 ON THE Y-AXIS. THE POINTS (1, −1), (2, 5), (3, −2), (1, 1), (4, 3)" class="img-fluid atto_image_button_text-bottom" height="347" src="https://k12.openstax.org/contents/raise/resources/ee925e215fe7152df93361e3c7ca8524294e4166" width="350"></p>
<li>What do you notice about the points \((1, -1)\) and \((1,1)\)?</li>
<p> <strong>Answer: </strong> They are on the same vertical line.</p>
</ol>
<p class="os-raise-text-bold">VERTICAL LINE TEST</p>
<p>A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the
graph in at most one point.</p>
<p>If any vertical line intersects the graph in more than one point, the graph does not represent a function.</p>
<p>If the graph does represent a function, we say it “passes the vertical line test.”</p>
<ol class="os-raise-noindent" start="4" type="1">
<li> For the function graphed below, determine if the graph passes the vertical line test and then explain how this identifies the graph as a function or not. </li>
</ol>
<img alt="The figure has two graphs. In graph a there is a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 2), (3, 0), and (6, negative 2). In graph b there is a parabola opening to the right graphed on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The parabola goes through the points (negative 1, 0), (0, 1), (0, negative 1), (3, 2), and (3, negative 2)." src="https://k12.openstax.org/contents/raise/resources/a2975dff9362ab8a21bc861a0065c6d04dff3045" width="300">
<ol class="os-raise-noindent" start="5" type="1">
<li> For the function graphed below, determine if the graph passes the vertical line test and then explain how this identifies the graph as a function or not. </li>
</ol>
<img alt="The figure has two graphs. In graph a there is a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 2), (3, 0), and (6, negative 2). In graph b there is a parabola opening to the right graphed on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The parabola goes through the points (negative 1, 0), (0, 1), (0, negative 1), (3, 2), and (3, negative 2)." src="https://k12.openstax.org/contents/raise/resources/cec1f2a3a759e96feeab8d3d55ec7730058d5bb2" width="300">
<p>Notice that the graph of a line will always be a function. This is called a <strong>linear function</strong>.</p>
<p class="os-raise-text-bold">Answer: </p>
<p><strong>VERTICAL LINE TEST</strong><br>
A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the
graph in at most one point.</p>
<p>If any vertical line intersects the graph in more than one point, the graph does not represent a function.</p>
<p>If the graph does represent a function, we say it “passes the vertical line test.”</p>
<ol class="os-raise-noindent" start="4" type="1">
<li>Yes, this passes the vertical line test (see the sample blue lines below) for all points on the line so the graph is a function.</li>
<p><img alt="The figure has a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 2), (3, 0), and (6, negative 2). Three dashed vertical straight lines are drawn at x equalsnegative 5, x equalsnegative 3, and x equals3. Each line intersects the slanted line at exactly one point." class="atto_image_button_text-bottom" height="325" src="https://k12.openstax.org/contents/raise/resources/4463db3a1f658e1734cf80330610c7e00c842471" width="317"></p>
<li> No, this graph does not pass the vertical line test, so it is not a function. For example, the vertical line \(x=1\) crosses the graph more than once.</li>
</ol><img alt="The figure has a parabola opening to the right graphed on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The parabola goes through the points (negative 1, 0), (0, 1), (0, negative 1), (3, 2), and (3, negative 2). Three dashed vertical straight lines are drawn at x equalsnegative 2, x equalsnegative 1, and x equals2. The vertical line x – negative 2 does not intersect the parabola. The vertical line x equalsnegative 1 intersects the parabola at exactly one point. The vertical line x equals3 intersects the parabola at two separate points." class="atto_image_button_text-bottom" height="311" src="https://k12.openstax.org/contents/raise/resources/46678dbea19435947004bff8050dd8ce0fbebcc2" width="305">
<p>Notice that the graph of a non-vertical line will always be a function. This is called a <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">llinear function. </span>. </p>
<h4>Video: Interpreting Function Notation </h4>
<p>Watch the following video to learn more about function notation.</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/fe84d773f2ac6fbbb436b9cc3b52bf2479c62f05">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/04c3307620f6a76063ddbcb4de503b09273a447a" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/fe84d773f2ac6fbbb436b9cc3b52bf2479c62f05
</video></div>
</div>
<br>
<br>
<h4>4.5.2: Self Check </h4>
<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>Which of the following graphs is the graph of a function?
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<img alt="The figure has a circle graphed on the x y-coordinate plane. The \(x\)-axis runs from negative 6 to 6. The \(y\)-axis runs from negative 6 to 6. The circle goes through the points (negative 3, 0), (3, 0), (0, negative 3), and (0, 3)." class="img-fluid atto_image_button_text-bottom" height="235" src="https://k12.openstax.org/contents/raise/resources/af8e3793684afca4a6c89f1a2c546a8fcd294907" width="229">
</td>
<td>
Incorrect. Let’s try again a different way: For a graph to be that of a function, it must pass the
vertical line test. For this graph, there are multiple places where an imaginary vertical line would touch
more than once. For example, the vertical line \(x = 2\) crosses the graph more than once. The answer is:<br>
<img alt="The figure has a parabola opening up graphed on the x y-coordinate plane. The \(x\)-axis runs from negative 6 to 6. The \(y\)-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 6), (1, 3), (0, 2), (1, 3), and (2, 6)." class="img-fluid atto_image_button_text-bottom" height="235" src="https://k12.openstax.org/contents/raise/resources/a334a980171d2cb3f7ed2a3be3d17e4c6ccfb0a6" width="229">
</td>
</tr>
<tr>
<td>
<img alt="The figure has a parabola opening right graphed on the x y-coordinate plane. The \(x\)-axis runs from negative 6 to 6. The \(y\)-axis runs from negative 6 to 6. The parabola goes through the points (negative 2, 0), (negative 1, 1), (negative 1, negative 1), (negative 2, 2), and (2, 2)." class="img-fluid atto_image_button_text-bottom" height="235" src="https://k12.openstax.org/contents/raise/resources/8bcdce1d348b87577df82f0bf6cad83db03dd7e9" width="229">
</td>
<td>
Incorrect. Let’s try again a different way: For a graph to be that of a function, it must pass the
vertical line test. For this graph, there are multiple places where an imaginary vertical line would touch
more than once. For example, the vertical line \(x =2\) crosses the graph more than once. The answer is:<br>
<img alt="The figure has a parabola opening up graphed on the x y-coordinate plane. The \(x\)-axis runs from negative 6 to 6. The \(y\)-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 6), (1, 3), (0, 2), (1, 3), and (2, 6)." class="img-fluid atto_image_button_text-bottom" height="235" src="https://k12.openstax.org/contents/raise/resources/a334a980171d2cb3f7ed2a3be3d17e4c6ccfb0a6" width="229">
</td>
</tr>
<tr>
<td>
<img alt="The figure has a parabola opening up graphed on the x y-coordinate plane. The \(x\)-axis runs from negative 6 to 6. The \(y\)-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 6), (1, 3), (0, 2), (1, 3), and (2, 6)." class="img-fluid atto_image_button_text-bottom" height="235" src="https://k12.openstax.org/contents/raise/resources/a334a980171d2cb3f7ed2a3be3d17e4c6ccfb0a6" width="229">
</td>
<td>
That’s correct! Check yourself: If an imaginary vertical line were drawn anywhere on the graph, it
would only touch once. This graph passes the vertical line test and is the graph of a function.
</td>
</tr>
<tr>
<td>
<img alt="The figure has a sideways absolute value function graphed on the x y-coordinate plane. The \(x\)-axis runs from negative 6 to 6. The \(y\)-axis runs from negative 6 to 6. The line bends at the point (0, 2) and goes to the right. The line goes through the points (1, 3), (2, 4), (1, 1), and (2, 0)." class="img-fluid atto_image_button_text-bottom" height="235" src="https://k12.openstax.org/contents/raise/resources/450c9e05950d38bccf933c33f97b7e287471c583" width="229">
</td>
<td>
Incorrect. Let’s try again a different way: For a graph to be that of a function, it must pass the
vertical line test. For this graph, there are multiple places where an imaginary vertical line would touch
more than once. For example, the vertical line \(x = 2\) crosses the graph more than once. The answer is:<br>
<img alt="The figure has a parabola opening up graphed on the x y-coordinate plane. The \(x\)-axis runs from negative 6 to 6. The \(y\)-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 6), (1, 3), (0, 2), (1, 3), and (2, 6)." class="img-fluid atto_image_button_text-bottom" height="235" src="https://k12.openstax.org/contents/raise/resources/a334a980171d2cb3f7ed2a3be3d17e4c6ccfb0a6" width="229">
</td>
</tr>
</tbody>
</table>
<br>
<h4>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>Determine if a Graph Is a Function</h4>
<p><img alt="This figure has a graph next to a table. The graph has a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The line is labeled f of x equals2 x minus 3. There are several vertical arrows that relate values on the \(x\)-axis to points on the line. The first arrow relates x equalsnegative 2 on the \(x\)-axis to the point (negative 2, negative 7) on the line. The second arrow relates x equalsnegative 1 on the \(x\)-axis to the point (negative 1, negative 5) on the line. The next arrow relates x equals0 on the \(x\)-axis to the point (0, negative 3) on the line. The next arrow relates x equals3 on the \(x\)-axis to the point (3, 3) on the line. The last arrow relates x equals4 on the \(x\)-axis to the point (4, 5) on the line. The table has 7 rows and 3 columns. The first row is a title row with the label f of x equals2 x minus 3. The second row is a header row with the headers x, f of x, and (x, f of x). The third row has the coordinates negative 2, negative 7, and (negative 2, negative 7). The fourth row has the coordinates negative 1, negative 5, and (negative 1, negative 5). The fifth row has the coordinates 0, negative 3, and (0, negative 3). The sixth row has the coordinates 3, 3, and (3, 3). The seventh row has the coordinates 4, 5, and (4, 5)." class="img-fluid atto_image_button_text-bottom" height="469" src="https://k12.openstax.org/contents/raise/resources/8cafc286f38f1825dd57670ce189645f0e417f3d" width="651"> <br>
A relation is a function if every input has exactly one output value. So the relation defined by the equation
\(y=2x−3\) is a function. Notice that a line will always be a function, so it is called a linear function.</p>
<p>If we look at the graph, each vertical dashed line intersects the line at only one point. This makes sense because in
a function, for every \(x\)-value there is only one \(y\)-value.</p>
<p>If the vertical line hit the graph twice, the \(x\)-value would be mapped to two \(y\)-values, so the graph would not
represent a function.</p>
<p>This leads us to the vertical line test. A set of points in a rectangular coordinate system is the graph of a
function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph
in more than one point, the graph does not represent a function.</p>
<p>Look at the graphs below:<br>
<img alt="The figure has two graphs. In graph a there is a parabola opening up graphed on the x y-coordinate plane. The \(x\)-axis runs from negative 6 to 6. The \(y\)-axis runs from negative 2 to 10. The parabola goes through the points (0, negative 1), (negative 1, 0), (1, 0), (negative 2, 3), and (2, 3). In graph b there is a circle graphed on the x y-coordinate plane. The \(x\)-axis runs from negative 6 to 6. The \(y\)-axis runs from negative 6 to 6. The circle goes through the points (negative 2, 0), (2, 0), (0, negative 2), and (0, 2)." class="img-fluid atto_image_button_text-bottom" height="358" src="https://k12.openstax.org/contents/raise/resources/f036b97c1d7e062abba7b5b7eb4a54663d3d43f8" width="675">
</p>
<p>Graph a passes the vertical line test because if an imaginary vertical line were drawn anywhere on the graph, it
would only touch one time.</p>
<p>Graph a is a function.</p>
<p>Graph b does not pass the vertical line test because when a vertical line is drawn down the graph, there is at least
one place where the vertical line touches twice.</p>
<p>Graph b is not a function.</p>
<h4>Try It: Determine if a Graph Is a Function</h4>
<p>Determine if each graph is a function. Use the vertical line test to explain.</p>
<p><img alt="The figure has two graphs. In graph a there is an ellipse graphed on the x y-coordinate plane. The \(x\)-axis runs from negative 6 to 6. The \(y\)-axis runs from negative 6 to 6. The ellipse goes through the points (0, negative 3), (negative 2, 0), (2, 0), and (0, 3). In graph b there is a straight line graphed on the x y-coordinate plane. The \(x\)-axis runs from negative 12 to 12. The \(y\)-axis runs from negative 12 to 12. The line goes through the points (0, negative 2), (2, 0), and (4, 2)." class="img-fluid atto_image_button_text-bottom" height="359" src="https://k12.openstax.org/contents/raise/resources/5b86b8dab19477d50e868d209cf51562ebc30909" width="679"></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 determine if a graph is a function:</p>
<p>Graph a does not pass the vertical line test because if a vertical line were drawn down the graph, there would be at
least one place where the vertical line would touch more than once. For example, a vertical line drawn at \(x=1\)
would cross the graph more than once.</p>
<p>Graph a is not a function.</p>
<p>Graph b does pass the vertical line test since a vertical line drawn anywhere on the graph will only touch one time.
</p>
<p>Graph b is a function.</p>
</p>