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11.2: Optional section- Rational functions by hand

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    49018
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    In this section we will show how to sketch the graph of a factored rational function without the use of a calculator. It will be helpful to the reader to have read section 3 on graphing a polynomial by hand before continuing in this section. In addition to having the same difficulties as polynomials, calculators often have difficulty graphing rational functions near an asymptote.

    Example \(\PageIndex{1}\)

    Graph the function \(p(x)=\dfrac{-3x^2(x-2)^3(x+2)}{(x-1)(x+1)^2(x-3)^3}\).

    Solution

    We can see that \(p\) has zeros at \(x=0,2,\) and \(-2\) and vertical asymptotes \(x=1\), \(x=-1\) and \(x=3\). Also note that for large \(|x|\), \(p(x)\approx -3\). So there is a horizontal asymptote \(y=-3\). We indicate each of these facts on the graph:

    clipboard_e1ab5cc017086ae6ddfbc5b1e97a886ca.png

    We can in fact get a more precise statement by performing a long division and writing \(p(x)=\dfrac{n(x)}{d(x)}= -3+\dfrac{r(x)}{d(x)}\). If you drop all but the leading order terms in the numerator and the denominator of the second term, we see that \(p(x)\approx -3-\frac{12}{x}\) whose graph for large \(|x|\) looks like

    clipboard_ec38d2a7bd99fe6baace384781d7e058b.png

    This sort of reasoning can make the graph a little more accurate but is not necessary for a sketch.

    We also have the following table:

    \[\begin{array}{r|l|l|l}
    \text { for } a & \text { near } a, p(x) \approx & \text { type } & \text { sign change at } a \\
    \hline-2 & C_{1}(x+2) & \text { linear } & \text { changes } \\
    -1 & C_{2} /(x+1)^{2} & \text { asymptote } & \text { does not change } \\
    0 & C_{3} x^{2} & \text { parabola } & \text { does not change } \\
    1 & C_{4} /(x-1) & \text { asymptote } & \text { changes } \\
    2 & C_{5}(x-2)^{3} & \text { cubic } & \text { changes } \\
    3 & C_{6} /(x-3)^{3} & \text { asymptote } & \text { changes }
    \end{array} \nonumber \]

    Note that if the power appearing in the second column is even then the function does not change from one side of \(a\) to the other. If the power is odd, the sign changes (either from positive to negative or from negative to positive).

    Now we move from large negative \(x\) values toward the right, taking into account the above table. For large negative \(x\), we start our sketch as follows:

    clipboard_ec81bedade4b6eda8f6ade524b3e1e6af.png

    And noting that near \(x=-2\) the function \(p(x)\) is approximately linear, we have

    clipboard_e989926d8794a318a3cefea902bc0570c.png

    Then noting that we have an asymptote (noting that we can not cross the \(x\)-axis without creating an \(x\)-intercept) we have

    clipboard_e04a2ea647dacaef54850fb4154e2331c.png

    Now, from the table we see that there is no sign change at \(-1\) so we have

    clipboard_e9e714f40cc75354c84e362afd3b3a8dd.png

    and from the table we see that near \(x=0\) the function \(p(x)\) is approximately quadratic and therefore the graph looks like a parabola. This together with the fact that there is an asymptote at \(x=1\) gives

    clipboard_efc1a9a9350a0e803a3a050301a3b7811.png

    Now, from the table we see that the function changes sign at the asymptote, so while the graph “hugs” the top of the asymptote on the left hand side, it “hugs” the bottom on the right hand side giving

    clipboard_e0d63cea5abaf6524dc049fffda77d4b3.png

    Now, from the table we see that near \(x=2\), \(p(x)\) is approximately cubic. Also, there is an asymptote \(x=3\) so we get

    clipboard_e49f2ce397ee19fd103adfd949b53cbc3.png

    Finally, we see from the table that \(p(x)\) changes sign at the asymptote \(x=3\) and has a horizontal asymptote \(y=-3\), so we complete the sketch:

    clipboard_ed49fe558b902b221b968d4e526646c6b.png

    Note that if we had made a mistake somewhere there is a good chance that we would have not been able to get to the horizontal asymptote on the right side without creating an additional \(x\)-intercept.

    What can we conclude from this sketch? This sketch exhibits only the general shape which can help decide on an appropriate window if we want to investigate details using technology. Furthermore, we can infer where \(p(x)\) is positive and where \(p(x)\) is negative. However, it is important to notice, that there may be wiggles in the graph that we have not included in our sketch.

    We now give one more example of graphing a rational function where the horizontal asymptote is \(y=0\).

    Example \(\PageIndex{2}\)

    Sketch the graph of \[r(x)=\dfrac{2x^2(x-1)^3(x+2)}{(x+1)^4(x-2)^3} \nonumber \]

    Solution

    Here we see that there are \(x\)-intercepts at \((0,0)\), \((0,1)\), and \((0,-2)\). There are two vertical asymptotes: \(x=-1\) and \(x=2\). In addition, there is a horizontal asymptote at \(y=0\). (Why?) Putting this information on the graph gives

    clipboard_e4f859389afc0f55141632cd6fb48f000.png

    In this case, it is easy to get more information for large \(|x|\) that will be helpful in sketching the function. Indeed, when \(|x|\) is large, we can approximate \(r(x)\) by dropping all but the highest order term in the numerator and denominator which gives \(r(x)\approx\dfrac{2x^6}{x^7}=\frac{2}{x}\). So for large \(|x|\), the graph of \(r\) looks like

    clipboard_e6d8053242b9025eb3893ad0d3846e40f.png

    The function gives the following table:

    \[\begin{array}{r|l|l|l}
    \text { for } a & \text { near } a, p(x) \approx & \text { type } & \text { sign change at } a \\
    \hline-2 & C_{1}(x+2) & \text { linear } & \text { changes } \\
    -1 & C_{2} /(x+1)^{4} & \text { asymptote } & \text { does not change } \\
    0 & C_{3} x^{2} & \text { parabola } & \text { does not change } \\
    1 & C_{4}(x-1)^{3} & \text { cubic } & \text { changes } \\
    2 & C_{5} /(x-2)^{3} & \text { asymptote } & \text { changes }
    \end{array} \nonumber \]

    Looking at the table for this function, we see that the graph should look like a line near the zero \((0,-2)\) and since it has an asymptote \(x=-1\), the graph looks something like:

    clipboard_e3750fd92635a6d8bdc1c8744b304474c.png

    Then, looking at the table we see that \(r(x)\) does not change its sign near \(x=-1\), so that we obtain:

    clipboard_e02cc987f68f398981d1e61fea94e2212.png

    Now, the function is approximately quadratic near \(x=0\) so the graph looks like:

    clipboard_e1df40fffe76ea770a397737b676373c1.png

    Turning to head toward the root at \(1\) and noting that the function is approximately cubic there, and that there is an asymptote \(x=2\), we have:

    clipboard_e5a4f325fb48371afe146bc8194317af8.png

    Finally, we see that the function changes sign at \(x=2\) (see the table). So since the graph “hugs” the asymptote near the bottom of the graph on the left side of the asymptote, it will “hug” the asymptote near the top on the right side. So this together with the fact that \(y=0\) is an asymptote gives the sketch (perhaps using an eraser to match the part of the graph on the right that uses the large \(x\)):

    clipboard_e93dd677eac58d09310564e3ba5b17f65.png

    Note that if the graph couldn’t be matched at the end without creating an extra \(x\)-intercept, then a mistake has been made.


    This page titled 11.2: Optional section- Rational functions by hand is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.