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<title>How Polynomials Behave</title><!-- #EndEditable -->
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<h1 align="center">How Polynomials Behave</h1>
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<span class="larger">A <a href="polynomials.html">polynomial</a> looks like this:</span>
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<div class="beach">
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<table align="center" cellpadding="5" border="0">
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<tbody>
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<tr align="center">
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<td><img src="images/polynomial-1var-example.svg" alt="polynomial 2x^4+6x-5"></td>
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</tr>
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<tr align="center">
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<td>example of a polynomial</td>
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</tr>
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</tbody></table>
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</div>
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<h2>Continuous and Smooth</h2>
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<p>There are two main things about the graphs of Polynomials:</p>
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<div class="dotpoint">
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<p>The graphs of polynomials are <a href="../calculus/continuity.html">continuous</a>, which is a special term with an exact definition in calculus, but here we will use this simplified definition:</p>
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</div>
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<div class="center80">
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<p> <img src="../images/pencil-paper.gif" alt="pencil" style="float:left; margin: 0 10px 5px 0; vertical-align:middle;" height="77" width="91"><span class="larger">we can draw it without lifting our pen from the paper</span></p>
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<div style="clear:both"></div>
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</div>
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<div class="dotpoint">
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<p>The graphs of polynomials are also <b>smooth</b>. No sharp "corners" or "cusps"</p>
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<p><img src="images/polynomial-smooth.gif" alt="smooth, not sharp" height="113" width="231"></p>
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</div>
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<h2>How the Curves Behave</h2>
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<p>Let us graph some polynomials to see what happens ...</p>
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<p align="center">... and let us start with the simplest form:</p>
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<p class="largest" align="center">f(x) = x<sup>n</sup></p>
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<p>Which actually does interesting things.</p>
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<p class="center"><img src="images/power-functions-even.gif" alt="Even Power Functions" height="166" width="225"></p>
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<p><b>Even values</b> of "n" behave the same:
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</p>
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<ul>
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<li>Always above (or equal to) 0</li>
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<li>Always go through (0,0), (1,1) and (-1,1)</li>
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<li>Larger values of n flatten out near 0, and rise more sharply above the x-axis</li></ul>
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<p><br></p>
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<p>And:</p>
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<p class="center"><img src="images/power-functions-odd.gif" alt="Odd Power Functions" height="195" width="225"></p>
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<p><b>Odd values</b> of "n" behave the same
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</p>
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<ul>
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<li>Always go from negative <b>x</b> and <b>y</b> to positive <b>x</b> and <b>y</b></li>
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<li>Always go through (0,0), (1,1) and (−1,−1)</li>
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<li>Larger values of n flatten out near 0, and fall/rise more sharply from the x-axis</li></ul>
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<h2>Power Function of Degree n</h2>
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<p>Next, by including a multiplier of <span class="large">a</span> we get what is called a "Power Function":</p>
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<p align="center"><span class="largest">f(x) = ax<sup>n</sup></span><sup><br>
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</sup><span class="large">f(x) </span>equals<span class="large"> a</span> times <span class="large">x</span> to the "power" (ie exponent) <span class="large">n</span></p>
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<p>The "a" changes it this way:</p>
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<ul>
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<li>Larger values of <b>a</b> squash the curve (inwards to y-axis)</li>
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<li>Smaller values of <b>a</b> expand it (away from y-axis)</li>
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<li>And negative values of <b>a</b> flip it upside down</li>
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</ul>
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<table align="center" border="0">
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<tbody>
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<tr align="center">
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<td class="larger">Example: f(x) = ax<sup>2</sup><br>
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a = 2, 1, ½, −1</td>
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<td width="40"> </td>
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<td><span class="larger">Example: f(x) = ax<sup>3</sup><br>
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a = 2, 1, ½, −1</span></td>
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</tr>
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<tr align="center">
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<td> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td><img src="images/graph-ax2.gif" alt="ax^2" height="217" width="184"><br></td>
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<td> </td>
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<td><img src="images/graph-ax3.gif" alt="ax^3" height="217" width="185"></td>
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</tr>
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</tbody></table>
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<p>We can use that knowledge when sketching some polynomials:</p>
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<div class="example">
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<h3>Example: Make a Sketch of y=1−2x<sup>7</sup></h3>
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<p> Start with the simplest "odd power" graph of <span class="large">x<sup>3</sup></span>, and gradually turn it into <span class="large">1−2x<sup>7</sup></span></p>
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<ul>
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<li>We know how x<sup>3</sup> looks,</li>
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<li>x<sup>7</sup> is similar, but flatter near zero, and steeper elsewhere,</li>
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<li>Squash it to get 2x<sup>7</sup>,</li>
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<li>Flip it to get −2x<sup>7</sup>, and</li>
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<li>Raise it by 1 to get 1−2x<sup>7</sup>.</li>
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</ul>
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<p>Like this:</p>
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<p class="center"><img src="images/power-function-example1.gif" alt="x^3 to 1-2x^7 in steps" height="156" width="602"></p>
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<p>So by doing this step-by-step we can get a good result.</p>
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</div>
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<h2>Turning Points</h2>
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<p>A Turning Point is an x-value where a <a href="functions-maxima-minima.html">local maximum or local minimum</a> happens:</p>
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<p align="center"><img src="images/function-max-min.svg" alt="Local Max and Min"></p>
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<h3>How many turning points does a polynomial have?</h3>
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<p class="large" align="center">Never more than the Degree minus 1</p>
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<div class="words">
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<p>The <a href="degree-expression.html">Degree</a> of a Polynomial with one variable is <b> the <a href="../exponent.html">largest exponent</a> of that variable.</b></p>
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<p align="center"><img src="images/degree-example-a.svg" alt="polynomial"></p>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example: a polynomial of Degree 4 will have 3 turning points or less</h3>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td><img src="images/polynomial-turning-points1.gif" alt="x^4-2x^2+x" height="212" width="216"></td>
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<td> </td>
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<td><img src="images/polynomial-turning-points2.gif" alt="x^4-2x" height="215" width="215"></td>
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</tr>
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<tr align="center">
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<td><span class="large">x<sup>4</sup>−2x<sup>2</sup>+x<br>
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</span> has <b>3</b> turning points</td>
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<td> </td>
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<td><span class="large">x<sup>4</sup>−2x<br>
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</span> has only <b>1</b> turning point</td>
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</tr>
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</tbody></table>
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<br>
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<p>The <b>most</b> is 3, but there can be less.</p>
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</div>
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<p>We may not know where they are, but at least we know the most there can be!</p>
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<h2></h2>
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<h2>What Happens at the Ends</h2>
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<p>And when we move far from zero: </p>
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<ul>
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<li>far to the <b>right</b> (large values of x), or </li>
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<li>far to the <b>left</b> (large negative values of x) </li>
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</ul>
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<p>then the graph starts to resemble the graph of <b>y = ax<sup>n</sup></b> where <b>ax<sup>n</sup></b> is the <b>term with the highest degree</b>.</p>
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<div class="example">
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<h3>Example: f(x) = 3x<sup>3</sup>−4x<sup>2</sup>+x</h3>
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<p>Far to the left or right, the graph will look like <span class="large">3x<sup>3</sup></span></p>
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<table align="center" border="0">
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<tbody>
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<tr align="center">
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<td><img src="images/polynomial-end-behavior1.gif" alt="polynomial end behavior" height="141" width="189"></td>
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<td> </td>
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<td><img src="images/polynomial-end-behavior2.gif" alt="polynomial end behavior" height="142" width="143"></td>
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</tr>
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<tr align="center">
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<td>Near Zero, they are <br>
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different</td>
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<td> </td>
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<td>Far From Zero, they <br>
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become similar</td>
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</tr>
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</tbody></table>
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<p>This makes sense, because when x is large, then x<sup>3</sup> is much greater than x<sup>2</sup> etc</p>
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</div>
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<div class="words">
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<p>This is officially called the "<b>End Behavior Model</b>".</p>
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</div>
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<p>And yes, we have come to the end!</p>
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<h2>Summary</h2>
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<ul>
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<div class="bigul">
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<li>Graphs are <b>continuous</b> and <b>smooth</b></li>
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<li><b>Even</b> exponents behave the same: above (or equal to) 0; go through (0,0), (1,1) and (−1,1); larger values of n flatten out near 0, and rise more sharply.</li>
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<li><b>Odd</b> exponents behave the same: go from negative <b>x</b> and <b>y</b> to positive <b>x</b> and <b>y</b>;
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go through (0,0), (1,1) and (−1,−1); larger values of n flatten out near 0, and fall/rise more sharply</li>
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<li>Factors:
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<ul>
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<li>Larger values squash the curve (inwards to y-axis)</li>
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<li>Smaller values expand it (away from y-axis)</li>
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<li>And negative values flip it upside down</li>
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</ul>
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</li>
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<li>Turning points: there are "Degree − 1" or less.</li>
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<li>End Behavior: use the term with the largest exponent</li>
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</div>
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</ul>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(486, 487, 488, 4020, 1128, 4022, 1129, 2404, 2405, 4021);</script>
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<div class="related">
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<a href="introduction.html">Introduction to Algebra</a>
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<a href="definitions.html">Algebra - Basic Definitions</a>
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<a href="index.html">Algebra Index</a> </div>
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