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<h4>Activity (15 minutes)</h4>
<p>In this activity, students apply what they learned about the connections between quadratic expressions and the graphs
representing them. They also practice identifying equivalent quadratic expressions in standard and factored forms.
Students are given a set of cards containing equations and graphs. They sort them into sets of three cards wherein
each set contains two equivalent equations and a graph that all represent the same quadratic function. They also
explain to a partner how they know the cards belong together. A sorting task gives students opportunities to analyze
the different equations, graphs, and structures closely and make connections and to justify their decisions as they
practice constructing logical arguments. Here are the <a
href="https://k12.openstax.org/contents/raise/resources/e9f640491d13212b4dd964beb46b846a43335476" target="_blank">blackline
masters</a>.</p>
<p>Here are the equations and graphs for reference and planning:</p>
<p>\(y=x^2-1\)</p>
<p>\(y=x^2-4x\)</p>
<p>\(y=x^2-4x+4\)</p>
<p>\(y=x^2-5x+4\)</p>
<p>\(y=x(x-4)\)</p>
<p>\(y=(x+1)(x-1)\)</p>
<p>\(y=(x-1)(x-4)\)</p>
<p>\(y=(x-2)^2\)</p>
<p><img alt="Card sort images." height="344"
src="https://k12.openstax.org/contents/raise/resources/6e1588238e3876c5d84a23d9d31b742b47a4d202" width="624"></p>
<p>As students work, monitor how they go about making the matches. Some students may begin by studying features of the
graph and then relate them to the equations. Others may work the other way around. Identify students with contrasting
approaches so they can share in the discussion.</p>
<h4>Launch</h4>
<p>Arrange students in groups of two. Give each group a set of pre-printed cards. Tell students to take turns sorting
the cards into sets that represent the same quadratic function. The person whose turn it is to compile a set should
explain how they know the cards belong together. (The person who has the last turn should also explain why the cards
belong together, aside from the fact that they are the last remaining cards.) The partner should listen and ask for
clarification or discuss any disagreement. Once all the cards are sorted, ask students to record their findings in the
given graphic organizer.</p>
<p>If time permits, before partners record anything, ask them to compare their sorted sets with another group of
students and discuss any disagreements.</p>
<h4>Student Activity</h4>
<p>Your teacher will give your group a set of cards. Each card contains a graph or an equation.</p>
<ul class="os-raise-noindent">
<li> Take turns with your partner to sort the cards into sets so that each set contains two equations and a graph that
all represent the same quadratic function. </li>
<li> For each set of cards that you put together, explain to your partner how you know they belong together. </li>
<li> For each set that your partner puts together, listen carefully to their explanation. If you disagree, discuss
your thinking and work to reach an agreement. </li>
</ul>
<p>Once all the cards are sorted and discussed, record the equivalent equations, sketch the corresponding graph, and
write a brief note or explanation about why the representations were grouped together.</p>
<p>Write down your answers, then select the <strong>solution</strong> button to compare your work.</p>
<p class="os-raise-text-bold">Answers:</p>
<p>Set 1: <br>
Standard Form: \(y=x^2-1\) <br>
Factored Form: \(y=(x+1)(x-1)\)</p>
<p><img height="168" src="https://k12.openstax.org/contents/raise/resources/82fc8ea2f2005ffe3659eb6903b2408f9f0900db"
width="264"></p>
<p>Set 2:<br>
Standard Form: \(y=x^2-4x\) <br>
Factored Form: \(y=x(x-4)\)</p>
<p><img height="168" src="https://k12.openstax.org/contents/raise/resources/8d19fef8477df88c1a3420a749c54e64aff87df2"
width="264"></p>
<p>Set 3:<br>
Standard Form: \(y=x^2-5x+4\) <br>
Factored Form: \(y=(x-1)(x-4)\)</p>
<p><img height="168" src="https://k12.openstax.org/contents/raise/resources/71dd23549243c8d7e836891aaec751799982e634"
width="264"></p>
<p>Set 4:<br>
Standard Form: \(y=x^2-4x+4\) <br>
Factored Form: \(y=(x-2)^2\)</p>
<p><img height="192" src="https://k12.openstax.org/contents/raise/resources/b952e365f4dbda61bd3cbb0189fd5563b3a139b3"
width="264"></p>
<h4>Anticipated Misconceptions</h4>
<p>Some students may think a factor such as \((x-1)\) relates to an \(x\)-intercept of \((-1,0)\). In earlier lessons,
students learned that the zeros of the function give the \(x\)-coordinates of the \(x\)-intercepts. Show students the
equations \(x-1=0\) and \(x+1=0\) and ask them to solve each equation and relate the solutions back to making the
expression \((x+1)(x-1)\) equal 0. Some students may benefit from seeing the expression \(x-1\) written as \(x+-1\) to
further emphasize that the expression takes the value 0 when \(x\) is the opposite. Students will continue this work
when solving quadratic equations in the next unit.</p>
<h4>Activity Synthesis</h4>
<p>Invite students to share how they found pairs of equivalent equations and how they matched the equations to the
graphs.</p>
<p>Highlight these explanations:</p>
<ul class="os-raise-noindent">
<li> The \(y\)-intercept of the graph helps to find the equation in standard form (or vice versa). </li>
<li> The \(x\)-intercepts of the graph help find the equation in factored form (or vice versa). </li>
<li> The quadratic expression in factored form can be expanded to find the equivalent expression in standard form.
</li>
</ul>
<h3>7.12.4: 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>Examine the graph below.</p>
<p><img
alt="GRAPH OF A PARABOLA THAT OPENS UPWARD WITH A \(y\)-intercepts OF NEGATIVE 2 AND \(x\)-intercepts OF NEGATIVE 2 AND 1."
height="203" src="https://k12.openstax.org/contents/raise/resources/2a5ccb596e0b86a9c47f805dc960f21769cf705e"
width="213"></p>
<p>Which function matches the graph?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">
Answers
</th>
<th scope="col">
Feedback
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
\(f(x) = (x - 2)(x + 1)\)
</td>
<td>
Incorrect. Let’s try again a different way: The zeros for this function are 2 and -1. The \(x\)-values
that make the factors 0 are the zeros on the graph. The answer is \(f(x) = (x - 2)(x + 1)\).
</td>
</tr>
<tr>
<td>
\(f(x) = (x + 2)(x - 1)\)
</td>
<td>
That’s correct! Check yourself: The values \(x=-2\) and \(x=1\) will make each function equal 0. They are
also zeros on the graph.
</td>
</tr>
<tr>
<td>
\(f(x) = (x + 2)(x + 1)\)
</td>
<td>
Incorrect. Let’s try again a different way: The zeros for this function are -2 and -1. The \(x\)-values
that make the factors 0 are the zeros on the graph. The answer is \(f(x) = (x - 2)(x + 1)\).
</td>
</tr>
<tr>
<td>
\(f(x) = (x - 2)(x - 1)\)
</td>
<td>
Incorrect. Let’s try again a different way: The zeros for this function are 2 and 1. The \(x\)-values that
make the factors 0 are the zeros on the graph. The answer is \(f(x) = (x - 2)(x + 1)\).
</td>
</tr>
</tbody>
</table>
<br>
<h3>7.12.4: 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>Matching Quadratic Functions and Their Graphs</h4>
<p>Below is a quadratic function represented in standard form, factored form, and with a graph.</p>
<p>Standard Form: \(f(x)=x^2+2x-8\)<br>
Factored Form: \(f(x) = (x + 4)(x - 2)\) </p>
<p><img
alt="GRAPH OF A PARABOLA THAT OPENS UPWARD WITH A \(y\)-intercepts OF NEGATIVE 8 AND \(x\)-intercepts OF NEGATIVE 4 AND 2."
height="302" src="https://k12.openstax.org/contents/raise/resources/337840b5d9ad8ca8808c7a428009e9f710edb0dc"
width="300"></p>
<p>Notice the factored form helps find the zeros: \(x = -4\) and \(x = 2\).</p>
<p>The factored form gives the zeros since the zeros (\(x\)-intercepts) are located where each factor equals 0.</p>
<p>The standard form, \(f(x)=x^2+2x-8\), helps give the \(y\)-intercept. The \(y\)-intercept occurs when \(x = 0\).
Here, \(f(0) = -8\). On the graph, the \(y\)-intercept is \((0, -8)\).</p>
<h4>Try It: Matching Quadratic Functions and Their Graphs</h4>
<p>Given the graph below and its function in standard form and factored form, identify all intercepts.</p>
<p><img
alt="GRAPH OF A PARABOLA THAT OPENS UPWARD WITH A \(y\)-intercepts OF NEGATIVE 6 AND \(x\)-intercepts OF NEGATIVE 2 AND 3."
height="302" src="https://k12.openstax.org/contents/raise/resources/12de1fe6540ce362885b3f9b87affd281056860b"
width="300"></p>
<p>Standard form: \(f(x)=x^2-x-6\)<br>
Factored form: \(f(x) = (x - 3)(x + 2)\) </p>
<p>Write down your answers, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer: </strong>Here is how to find the intercepts with the information provided: </p>
<p>The graph and the factored form help give the \(x\)-intercepts. They occur where the factors of the factored form
equal 0, at \(x = -2\) and \(x = 3\). On the graph, these are where the graph crosses the \(x\)-axis, at the points
\((-2, 0)\) and \((3, 0)\).</p>
<p>The \(y\)-intercept can be found with standard form when \(x=0\) in \(f(x)=x^2-x-6\). So \(y = -6\). <br>
Notice on the graph, the \(y\)-intercept, where the graph crosses the \(y\)-axis, is at \((0, -6)\).</p>