Write a rational function given intercepts and asymptotes. Review: What Are Rational Functions? To sketch a graph of a rational function, you can start by finding the asymptotes and intercepts. [latex]\begin{cases}-2=a\frac{\left(0+2\right)\left(0 - 3\right)}{\left(0+1\right){\left(0 - 2\right)}^{2}}\hfill \\ -2=a\frac{-6}{4}\hfill \\ \text{ }a=\frac{-8}{-6}=\frac{4}{3}\hfill \end{cases}[/latex], http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175. Find more Mathematics widgets in Wolfram|Alpha. See . Explore math with our beautiful, free online graphing calculator. To find the vertical asymptotes, we determine when the denominator is equal to zero. Lets do RAHTEY on it so we can extract a little more about the function’s orientation. (An exception occurs in the case of a removable discontinuity.) Given the function [latex]f\left(x\right)=\frac{{\left(x+2\right)}^{2}\left(x - 2\right)}{2{\left(x - 1\right)}^{2}\left(x - 3\right)}[/latex], use the characteristics of polynomials and rational functions to describe its behavior and sketch the function. Now some textbooks will have a different order or procedure for graphing Rational Functions than what you will see in this video, but what is important to note is that as long as we identify each and every essential element it doesn’t matter what order or steps you use…. Thanks to all of you who support me on Patreon. Definition of a Rational Function A rational function f has the form f(x) = g(x) h(x) where g(x) and h(x) are polynomial functions. Horizontal asymptote at [latex]y=\frac{1}{2}[/latex]. At the vertical asymptote [latex]x=2[/latex], corresponding to the [latex]\left(x - 2\right)[/latex] factor of the denominator, the graph heads towards positive infinity on the left side of the asymptote and towards negative infinity on the right side, consistent with the behavior of the function [latex]f\left(x\right)=\frac{1}{x}[/latex]. As with polynomials, factors of the numerator may have integer powers greater than one. Now that we have analyzed the equations for rational functions and how they relate to a graph of the function, we can use information given by a graph to write the function. At the [latex]x[/latex]-intercept [latex]x=3[/latex] corresponding to the [latex]\left(x - 3\right)[/latex] factor of the numerator, the graph passes through the axis as we would expect from a linear factor. The graph of a rational function usually has vertical asymptotes where the denominator equals 0. 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