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<h4> Evaluating a Polynomial Function for a Given Value</h4>
<p>A <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">polynomial function</span> is a function whose range values are defined by a polynomial. For example, \(f(x) = x^2 + 5x + 6\) and \(g(x) = 3x-4\) are polynomial functions because \(x^2 + 5x + 6\) and \(3x - 4\) are polynomials.</p>
<p>To evaluate a polynomial function, we will substitute the given value for the variable and then simplify using the order of operations.</p>
<p>Let’s look at some examples.</p>
<p>For the function \(f(x) = 5x^2 - 8x + 4\), find:</p>
<ol class="os-raise-noindent">
<li>\( f(4)\) </li>
<li> \(f(-2)\) </li>
<li> \(f(0)\) </li>
</ol>
<ol class="os-raise-noindent">
<p>The following is a step-by-step breakdown.
</p>
<li>\(f(x)=5x^2-8x+4\)</li>
</ol>
<p><strong>Step 1 -</strong> To find \(f(4)\), substitute \({\style{color:red}4}\) for \(x\). </p>
<p> \(f({\style{color:red}4})=5({\style{color:red}4})^2-8({\style{color:red}4})+4\) </p>
<p><strong>Step 2 -</strong> Simplify the exponents.</p>
<p> \(f(4)=5 \cdot 16-8(4)+4\)</p>
<p><strong>Step 3 -</strong> Multiply.</p>
<p> \(f(4)=80-32+4\)</p>
<p><strong>Step 4 -
</strong>Simplify.</p>
<p> \(f(4)=52\)</p>
<br>
<ol class="os-raise-noindent" start="2">
<li>\(f(x)=5x^2-8x+4\)</li>
</ol>
<p><strong>Step 1 -
</strong>To find \(f(-2)\), substitute \({\style{color:red}-}{\style{color:red}2}\) for \(x\).</p>
<p> \(f({\style{color:red}-}{\style{color:red}2})=5({\style{color:red}-}{\style{color:red}2})^2-8({\style{color:red}-}{\style{color:red}2})+4\)</p>
<p><strong>Step 2 -
</strong>Simplify the exponents.</p>
<p> \(f(-2)=5 \cdot 4-8(-2)+4\)</p>
<p><strong>Step 3 -</strong> Multiply.</p>
<p> \(f(-2)=20+15+4\)</p>
<p><strong>Step 4 -</strong> Simplify.</p>
<p> \(f(-2)=40\)</p>
<br>
<ol class="os-raise-noindent" start="3">
<li>\(f(x)=5x^2-8x+4\)</li>
</ol>
<p><strong>Step 1 - </strong> To find \(f(0)\), substitute \({\style{color:red}0}\) for \(x\). </p>
<p> \(f({\style{color:red}0})=5({\style{color:red}0})^2-8({\style{color:red}0})+4\) </p>
<p><strong>Step 2 - </strong> Simplify the exponents.</p>
<p> \(f(0)=5 \cdot 0-8(0)+4\)</p>
<p><strong>Step 3 - </strong> Multiply.</p>
<p> \(f(0)=0+0+4\)</p>
<p><strong>Step 4 - </strong> Simplify.</p>
<p> \(f(0)=4\)</p>
<br>
<h4> Try It: Evaluating a Polynomial Function for a Given Value</h4>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<p>For the function \(f(x) = 3x^2 + 2x - 15\), find:</p>
<ol class="os-raise-noindent">
<li>\( f(3)\) </li>
<li> \(f(-5) \)</li>
<li> \(f(0) \)</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answers. Then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Here is how to evaluate the polynomial function:</p>
<ol class="os-raise-noindent">
<li>\( f(x) = 3x^2 + 2x - 15 \)<br>
\(f(3) = 3(3)^2 + 2(3) - 15\)<br>
\(f(3) = 3(9) + 6 - 15\)<br>
\(f(3) = 18\)</li>
</ol>
<ol class="os-raise-noindent" start="2">
<li>\( f(x) = 3x^2 + 2x - 15\)<br>
\(f(-5) = 3(-5)^2 + 2(-5) - 15\)<br>
\(f(-5) = 3(25) - 10 - 15\)<br>
\(f(-5) = 50\)</li>
</ol>
<ol class="os-raise-noindent" start="3">
<li>\( f(x) = 3x^2 + 2x - 15\)<br>
\(
f(0) = 3(0)^2 + 2(0) - 15\)<br>
\(
f(0) = -15\)</li>
</ol>
</div>