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05ef8702-1f4c-43f6-a045-de508ef5c23f.html
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<h4>Activity</h4>
<p>When finding an exact function that has particular zeros and passes through a certain point, we use the intercept form of a quadratic function:</p>
<p>\(f(x) = a(x − p)(x − q)\)</p>
<p>The variables \(p\) and \(q\) represent the zeros or roots of the function. The variable \(a\) accounts for any constant that might have been factored out when solving.</p>
<p>Remember, if needed in the following problems, use the "^" symbol to enter exponents.</p>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="90790d83-3288-4077-bd3b-dff4974d3b12" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li>Find a function, \(m(x)\), in intercept form, that has the zeros –5 and 8.</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter a function, \(m(x)\), in intercept form.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>The intercept form is \(m(x) = a(x − (-5))(x − 8)\). </p>
<p class="os-raise-noindent">\(= a(x + 5)(x − 8)\) </p>
</div>
</div>
<!--Interaction End -->
<br>
<p>Let's use the intercept form of the function above to find the exact function that passes through \((5, -60)\).</p>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="d0b152af-b2f1-43de-a14e-9b2fed58450d" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li>First, find the value of \(a\). To do this, substitute the values of the point into the intercept form of the function.</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter the value of \(a\).</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>\(m(x) = a(x + 5)(x − 8)\) <br>
\(-60 = a((5) + 5)((5) − 8)\) <br>
\(-60 = a(10)(-3)\) <br>
\(-60 = -30a\)<br>
\(2 = a\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="ab89d881-df9a-4c48-bfbb-8d7c5b34f7b7" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li>Substitute the value of \(a\) into the intercept form of the function and multiply to find the standard form.</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter the standard form of the function.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p> \(m(x) = a(x + 5)(x − 8)\)<br>
\(m(x) = 2(x + 5)(x − 8)\)<br>
\(m(x) = 2(x^2 + 5x − 8x − 40)\)<br>
\(m(x) = 2(x^2 − 3x − 40)\)<br>
\(m(x) = 2x^2 − 6x − 80\)</p>
<p>This is the function with zeros –5 and 8 that passes through \((5, -60)\).</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="a1f840d2-22c7-4db6-a338-049576b27769" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="4">
<li>Write the quadratic function, \(n(x)\), in standard form, that has the zeros 9 and –2 and passes through the point \((0, -36)\).</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter the quadratic function, \(n(x)\), in standard form.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>\(n(x) = 2x^2 − 14x − 36\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="3cbc8ecd-ad81-4836-a84a-f544485c9fcd" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="5">
<li>Write the quadratic function, \(p(x)\), in standard form, that has the zeros –6 and –4 and passes through the point \((-5, 3)\).</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter the quadratic function, \(p(x)\), in standard form.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>\(p(x) = −3x^2 − 30x − 72\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="e0050a30-ce71-4cfc-8b61-e7a9414122d2" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6">
<li>Write the quadratic function, \(c(x)\), in standard form, that has the zeros 1 and 7 and passes through the point \((9, 16)\).</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter the quadratic function, \(c(x)\), in standard form.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>\(c(x) = x^2 − 8x + 7\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<h4>Video: Finding a Quadratic Function from Its Zeros and a Point</h4>
<p>Watch the following video to learn more about finding a quadratic function from its zeros and a point. </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/edd0a3e9d4afdfbd30566216317a5425c3ae450b">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/9ec7085ed4a5348ba5bda6db2a8c2f66caeb427e" srclang="en_us">https://k12.openstax.org/contents/raise/resources/edd0a3e9d4afdfbd30566216317a5425c3ae450b
</video></div>
</div>