{"id":1014,"date":"2025-07-21T19:16:03","date_gmt":"2025-07-21T19:16:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1014"},"modified":"2026-01-14T18:48:55","modified_gmt":"2026-01-14T18:48:55","slug":"rational-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rational-functions-learn-it-3\/","title":{"raw":"Rational Functions: Learn It 4","rendered":"Rational Functions: Learn It 4"},"content":{"raw":"

Horizontal and Slant Asymptotes<\/h2>\r\nWhile vertical asymptotes describe the behavior of a graph as the output<\/em> gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input<\/em> gets very large or very small. Recall that a polynomial\u2019s end behavior will mirror that of the leading term. Likewise, a rational function\u2019s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.\r\n\r\n
We have learned that knowing the leading coefficient and degree of a polynomial function is useful when predicting its end behavior.\r\n[latex]\\\\[\/latex]\r\nIf the leading term is positive or negative, and has even or odd degree, it will tell us the toolkit function's graph behavior it will mimic: [latex]f(x)=x^2, \\quad f(x)=-x^2,\\quad f(x)=x^3,\\quad[\/latex] or [latex]\\quad f(x)=-x^3[\/latex].\r\n[latex]\\\\[\/latex]\r\nThe same idea applies to the ratio of leading terms\u00a0<\/em>of a\u00a0rational function.<\/section>There are three distinct outcomes when checking for horizontal asymptotes:\r\n
    \r\n \t
  • Case 1:<\/strong> If the degree of the denominator [latex]>[\/latex] degree of the numerator, there is a horizontal asymptote<\/strong> at [latex]y=0[\/latex].\r\n
    \r\n

    Example: [latex]f\\left(x\\right)=\\dfrac{4x+2}{{x}^{2}+4x - 5}[\/latex]<\/p>\r\nIn this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{4x}{{x}^{2}}=\\frac{4}{x}[\/latex]. This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\\left(x\\right)=\\frac{4}{x}[\/latex], and the outputs will approach zero, resulting in a horizontal asymptote at [latex]y=0[\/latex]. Note that this graph crosses the horizontal asymptote.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]\"Graph Horizontal Asymptote [latex]y=0[\/latex] when [latex]f\\left(x\\right)=\\dfrac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne{0}\\text{ where degree of }p<\\text{degree of q}[\/latex].[\/caption]<\/section><\/li>\r\n \t

  • Case 2:<\/strong> If the degree of the denominator [latex]<[\/latex] degree of the numerator by one, we get a slant asymptote.\r\n
    \r\n

    Example: [latex]f\\left(x\\right)=\\dfrac{3{x}^{2}-2x+1}{x - 1}[\/latex]<\/p>\r\nIn this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{3{x}^{2}}{x}=3x[\/latex]. This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\\left(x\\right)=3x[\/latex]. As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of [latex]g\\left(x\\right)=3x[\/latex] looks like a diagonal line, and since [latex]f[\/latex] will behave similarly to [latex]g[\/latex], it will approach a line close to [latex]y=3x[\/latex]. This line is a slant asymptote (NOTE: the graph of the function itself is just a \"sketch\" within the parameters of the asymptotes).\r\n

    <\/div>\r\n[caption id=\"attachment_4477\" align=\"aligncenter\" width=\"489\"]\"Graph Slant Asymptote when [latex]f\\left(x\\right)=\\dfrac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne 0[\/latex] where degree of [latex]p>\\text{ degree of }q\\text{ by }1[\/latex].[\/caption]<\/section>
    To find the equation of the slant asymptote, we'll need to divide the rational expression. The asymptote is the\u00a0quotient<\/strong> (or result of the division without the remainder).<\/section><\/li>\r\n \t
  • Case 3:<\/strong> If the degree of the denominator [latex]=[\/latex] degree of the numerator, there is a horizontal asymptote at [latex]y=\\frac{{a}_{n}}{{b}_{n}}[\/latex], where [latex]{a}_{n}[\/latex] and [latex]{b}_{n}[\/latex] are the leading coefficients of [latex]p\\left(x\\right)[\/latex] and [latex]q\\left(x\\right)[\/latex] for [latex]f\\left(x\\right)=\\frac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne 0[\/latex].\r\n
    \r\n

    Example: [latex]f\\left(x\\right)=\\dfrac{3{x}^{2}+2}{{x}^{2}+4x - 5}[\/latex]<\/p>\r\n

    In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{3{x}^{2}}{{x}^{2}}=3[\/latex]. This tells us that as the inputs grow large, this function will behave like the function [latex]g\\left(x\\right)=3[\/latex], which is a horizontal line. As [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to 3[\/latex], resulting in a horizontal asymptote at [latex]y=3[\/latex]. Note that this graph crosses the horizontal asymptote.<\/p>\r\n\r\n[caption id=\"attachment_4478\" align=\"aligncenter\" width=\"731\"]\"Graph Horizontal Asymptote when [latex]f\\left(x\\right)=\\frac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne 0\\text{ where degree of }p=\\text{degree of }q[\/latex].[\/caption]<\/section><\/li>\r\n<\/ul>\r\n

    Notice that, while the graph of a rational function will never cross a vertical asymptote<\/strong>, the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.<\/section>
    \r\n

    horizontal asymptotes of rational functions<\/h3>\r\nThe horizontal asymptote<\/strong> of a rational function can be determined by looking at the degrees of the numerator and denominator.\r\n
      \r\n \t
    • Case 1:<\/strong> Degree of numerator is less than<\/em> degree of denominator: horizontal asymptote at\u00a0[latex]y=0[\/latex]<\/li>\r\n \t
    • Case 2<\/strong>: Degree of numerator is greater than degree of denominator by one<\/em>: no horizontal asymptote; slant asymptote.\r\n
        \r\n \t
      • If the degree of the numerator is greater than the degree of the denominator by\u00a0more than one<\/em>, the end behavior of the function's graph will mimic that of the graph of the reduced ratio of leading terms.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
      • Case 3<\/strong>: Degree of numerator is equal to<\/em> degree of denominator: horizontal asymptote at ratio of leading coefficients.<\/li>\r\n<\/ul>\r\n<\/section>It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior<\/strong> of the graph will mimic the behavior of the reduced end behavior fraction.\r\n\r\n
        You'll need to graph the rational function [latex]f\\left(x\\right)\\ =\\ \\dfrac{\\left(4x^a+2\\right)}{\\left(x^b+2x-3\\right)}[\/latex] using an online graphing calculator. Adjust the [latex]a[\/latex] and the\u00a0[latex]b[\/latex] values to obtain graphs with the following horizontal asymptotes:\r\n
          \r\n \t
        1. Find values of [latex]a[\/latex] and [latex]b[\/latex] that will give\u00a0a graph with a slant asymptote.<\/li>\r\n \t
        2. Find values of [latex]a[\/latex] and [latex]b[\/latex]\u00a0that will give a graph with a horizontal asymptote at [latex]y = 0[\/latex]<\/li>\r\n \t
        3. Find values of [latex]a[\/latex] and [latex]b[\/latex] that will give a graph with a horizontal asymptote that is the ratio of the leading coefficients of [latex]f(x)[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"918195\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"918195\"]\r\n\r\nAnswers will vary.\r\n
            \r\n \t
          1. Find values of [latex]a[\/latex] and [latex]b[\/latex]\u00a0that will give\u00a0a graph with a slant asymptote. \u00a0For example [latex]b = 3, a = 4[\/latex]<\/li>\r\n \t
          2. Find values of [latex]a[\/latex] and [latex]b[\/latex] that will give a graph with a horizontal asymptote at [latex]y = 0[\/latex]. \u00a0For example, [latex]b = 4, a = 2[\/latex]<\/li>\r\n \t
          3. Find values of [latex]a[\/latex] and [latex]b[\/latex] that will give a graph with a horizontal asymptote that is the ratio of the leading coefficients of [latex]f(x)[\/latex].For example, [latex]b = 4, a = 4[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
            For the functions below, identify the horizontal or slant asymptote.\r\n
              \r\n \t
            1. [latex]g\\left(x\\right)=\\dfrac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[\/latex]<\/li>\r\n \t
            2. [latex]h\\left(x\\right)=\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex]<\/li>\r\n \t
            3. [latex]k\\left(x\\right)=\\dfrac{{x}^{2}+4x}{{x}^{3}-8}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"968793\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"968793\"]\r\n\r\nFor these solutions, we will use [latex]f\\left(x\\right)=\\dfrac{p\\left(x\\right)}{q\\left(x\\right)}, q\\left(x\\right)\\ne 0[\/latex].\r\n
                \r\n \t
              1. [latex]g\\left(x\\right)=\\dfrac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[\/latex]: The degree of [latex]p[\/latex] and the degree of [latex]q[\/latex] are both equal to 3, so we can find the horizontal asymptote by taking the ratio of the leading terms.\r\nThere is a horizontal asymptote at [latex]y=\\frac{6}{2}[\/latex] or [latex]y=3[\/latex].<\/li>\r\n \t
              2. [latex]h\\left(x\\right)=\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex]: The degree of [latex]p=2[\/latex] and degree of [latex]q=1[\/latex]. Since [latex]p>q[\/latex] by 1, there is no horizontal asymptote.\r\nHowever, we have a slant asymptote found at [latex]\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex].\r\n\"Synthetic\r\nThe quotient is [latex]x - 6[\/latex] and the remainder is [latex]13[\/latex]. There is a slant asymptote at [latex]y=-x - 6[\/latex].<\/li>\r\n \t
              3. [latex]k\\left(x\\right)=\\dfrac{{x}^{2}+4x}{{x}^{3}-8}[\/latex]: The degree of [latex]p=2\\text{ }<[\/latex] degree of [latex]q=3[\/latex], so there is a horizontal asymptote [latex]y=0[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
                Find the horizontal and vertical asymptotes of the function\r\n

                [latex]f\\left(x\\right)=\\dfrac{\\left(x - 2\\right)\\left(x+3\\right)}{\\left(x - 1\\right)\\left(x+2\\right)\\left(x - 5\\right)}[\/latex]<\/p>\r\n[reveal-answer q=\"228875\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"228875\"]\r\n\r\nFirst, note that this function has no common factors, so there are no potential removable discontinuities.\r\n\r\nThe function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The denominator will be zero at [latex]x=1,-2,\\text{and }5[\/latex], indicating vertical asymptotes at these values.<\/strong>\r\n\r\nThe numerator has degree 2, while the denominator has degree 3. Since the degree of the denominator is greater than the degree of the numerator, the denominator will grow faster than the numerator, causing the outputs to tend towards zero as the inputs get large, and so as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to 0[\/latex]. This function will have a horizontal asymptote at [latex]y=0[\/latex].<\/strong>\r\n\r\nBelow is the graph of the function to confirm our findings.\r\n\r\n\"Graph\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

                [ohm2_question hide_question_numbers=1]24655[\/ohm2_question]<\/section>
                [ohm_question hide_question_numbers=1]318944[\/ohm_question]<\/section>
                [ohm_question hide_question_numbers=1]318948[\/ohm_question]<\/section>","rendered":"

                Horizontal and Slant Asymptotes<\/h2>\n

                While vertical asymptotes describe the behavior of a graph as the output<\/em> gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input<\/em> gets very large or very small. Recall that a polynomial\u2019s end behavior will mirror that of the leading term. Likewise, a rational function\u2019s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.<\/p>\n

                We have learned that knowing the leading coefficient and degree of a polynomial function is useful when predicting its end behavior.
                \n[latex]\\\\[\/latex]
                \nIf the leading term is positive or negative, and has even or odd degree, it will tell us the toolkit function’s graph behavior it will mimic: [latex]f(x)=x^2, \\quad f(x)=-x^2,\\quad f(x)=x^3,\\quad[\/latex] or [latex]\\quad f(x)=-x^3[\/latex].
                \n[latex]\\\\[\/latex]
                \nThe same idea applies to the ratio of leading terms\u00a0<\/em>of a\u00a0rational function.<\/section>\n

                There are three distinct outcomes when checking for horizontal asymptotes:<\/p>\n

                  \n
                • Case 1:<\/strong> If the degree of the denominator [latex]>[\/latex] degree of the numerator, there is a horizontal asymptote<\/strong> at [latex]y=0[\/latex].
                  \n
                  \n

                  Example: [latex]f\\left(x\\right)=\\dfrac{4x+2}{{x}^{2}+4x - 5}[\/latex]<\/p>\n

                  In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{4x}{{x}^{2}}=\\frac{4}{x}[\/latex]. This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\\left(x\\right)=\\frac{4}{x}[\/latex], and the outputs will approach zero, resulting in a horizontal asymptote at [latex]y=0[\/latex]. Note that this graph crosses the horizontal asymptote.<\/p>\n

                  \"Graph
                  Horizontal Asymptote [latex]y=0[\/latex] when [latex]f\\left(x\\right)=\\dfrac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne{0}\\text{ where degree of }p<\\text{degree of q}[\/latex].<\/figcaption><\/figure>\n<\/section>\n<\/li>\n
                • Case 2:<\/strong> If the degree of the denominator [latex]<[\/latex] degree of the numerator by one, we get a slant asymptote.\n\n\n
                  \n

                  Example: [latex]f\\left(x\\right)=\\dfrac{3{x}^{2}-2x+1}{x - 1}[\/latex]<\/p>\n

                  In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{3{x}^{2}}{x}=3x[\/latex]. This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\\left(x\\right)=3x[\/latex]. As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of [latex]g\\left(x\\right)=3x[\/latex] looks like a diagonal line, and since [latex]f[\/latex] will behave similarly to [latex]g[\/latex], it will approach a line close to [latex]y=3x[\/latex]. This line is a slant asymptote (NOTE: the graph of the function itself is just a “sketch” within the parameters of the asymptotes).<\/p>\n

                  <\/div>\n
                  \"Graph
                  Slant Asymptote when [latex]f\\left(x\\right)=\\dfrac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne 0[\/latex] where degree of [latex]p>\\text{ degree of }q\\text{ by }1[\/latex].<\/figcaption><\/figure>\n<\/section>\n
                  To find the equation of the slant asymptote, we’ll need to divide the rational expression. The asymptote is the\u00a0quotient<\/strong> (or result of the division without the remainder).<\/section>\n<\/li>\n
                • Case 3:<\/strong> If the degree of the denominator [latex]=[\/latex] degree of the numerator, there is a horizontal asymptote at [latex]y=\\frac{{a}_{n}}{{b}_{n}}[\/latex], where [latex]{a}_{n}[\/latex] and [latex]{b}_{n}[\/latex] are the leading coefficients of [latex]p\\left(x\\right)[\/latex] and [latex]q\\left(x\\right)[\/latex] for [latex]f\\left(x\\right)=\\frac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne 0[\/latex].
                  \n
                  \n

                  Example: [latex]f\\left(x\\right)=\\dfrac{3{x}^{2}+2}{{x}^{2}+4x - 5}[\/latex]<\/p>\n

                  In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{3{x}^{2}}{{x}^{2}}=3[\/latex]. This tells us that as the inputs grow large, this function will behave like the function [latex]g\\left(x\\right)=3[\/latex], which is a horizontal line. As [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to 3[\/latex], resulting in a horizontal asymptote at [latex]y=3[\/latex]. Note that this graph crosses the horizontal asymptote.<\/p>\n

                  \"Graph
                  Horizontal Asymptote when [latex]f\\left(x\\right)=\\frac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne 0\\text{ where degree of }p=\\text{degree of }q[\/latex].<\/figcaption><\/figure>\n<\/section>\n<\/li>\n<\/ul>\n
                  Notice that, while the graph of a rational function will never cross a vertical asymptote<\/strong>, the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.<\/section>\n
                  \n

                  horizontal asymptotes of rational functions<\/h3>\n

                  The horizontal asymptote<\/strong> of a rational function can be determined by looking at the degrees of the numerator and denominator.<\/p>\n

                    \n
                  • Case 1:<\/strong> Degree of numerator is less than<\/em> degree of denominator: horizontal asymptote at\u00a0[latex]y=0[\/latex]<\/li>\n
                  • Case 2<\/strong>: Degree of numerator is greater than degree of denominator by one<\/em>: no horizontal asymptote; slant asymptote.\n
                      \n
                    • If the degree of the numerator is greater than the degree of the denominator by\u00a0more than one<\/em>, the end behavior of the function’s graph will mimic that of the graph of the reduced ratio of leading terms.<\/li>\n<\/ul>\n<\/li>\n
                    • Case 3<\/strong>: Degree of numerator is equal to<\/em> degree of denominator: horizontal asymptote at ratio of leading coefficients.<\/li>\n<\/ul>\n<\/section>\n

                      It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior<\/strong> of the graph will mimic the behavior of the reduced end behavior fraction.<\/p>\n

                      You’ll need to graph the rational function [latex]f\\left(x\\right)\\ =\\ \\dfrac{\\left(4x^a+2\\right)}{\\left(x^b+2x-3\\right)}[\/latex] using an online graphing calculator. Adjust the [latex]a[\/latex] and the\u00a0[latex]b[\/latex] values to obtain graphs with the following horizontal asymptotes:<\/p>\n
                        \n
                      1. Find values of [latex]a[\/latex] and [latex]b[\/latex] that will give\u00a0a graph with a slant asymptote.<\/li>\n
                      2. Find values of [latex]a[\/latex] and [latex]b[\/latex]\u00a0that will give a graph with a horizontal asymptote at [latex]y = 0[\/latex]<\/li>\n
                      3. Find values of [latex]a[\/latex] and [latex]b[\/latex] that will give a graph with a horizontal asymptote that is the ratio of the leading coefficients of [latex]f(x)[\/latex].<\/li>\n<\/ol>\n