\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Precalculus - Graphing Rational Functions

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

Graphing Rational Functions

Use the concepts of asymptotes and holes and then plot a few points to get a rough idea of what the function looks like. The interesting points are where the denominator is zero. The rest of a rational function is pretty smooth and easy to graph. Remember: If you use your calculator or graphing utility to check your answer, look at the graph with a critical eye and double-check everything since some details may be missed by your calculator.

1. If the highest power in the numerator is greater than or equal to the highest power in the denominator, you have a horizontal or slant asymptote. Use the techniques on the asymptotes page to extract the asymptote and the remaining rational function. Now work only with the remaining rational function.
2. Determine the domain of the function. Remember, this is where the denominator is not zero.
3. Determine if you have any vertical asymptotes or holes. The table, discussed in more detail on the basics of rational functions page, will help you.

\(r(x)\)

\(n(x) \neq 0\)

\(n(x) = 0\)

\(d(x) \neq 0\)

Rational Number

Zero

\(d(x) = 0\)

Vertical Asymptote

Hole

4. Now determine any zeroes of the function. The above table will help with finding where the graph crosses the x-axis. To determine where the graph crosses the y-axis, just plug in x = 0 if x = 0 is in the domain. If x = 0 is NOT in the domain, then the graph never crosses the y-axis.
5. Plot a few points to get a feel for what the graph looks like and fill in the remaining graph.

Okay, time for the practice problems.
Important Note - Some of the solution videos show the instructor plotting graphs using axes that are not labeled. Do not make that mistake in your work. It is very important to label the scales on your axes. As usual, check with your instructor to see what they require.

Practice

Sketch a graph of these rational functions by hand using the concepts of asymptotes, holes and zeroes discussed on this page.

\(\displaystyle{ f(x) = \frac{x^2}{x-2} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x^2}{x-2} }\)

Solution

2948 video solution

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\(\displaystyle{ f(x) = \frac{x^2-x-6}{x^2 + x - 2} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x^2-x-6}{x^2 + x - 2} }\)

Solution

2949 video solution

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\(\displaystyle{ f(x) = \frac{3x^2+4x-4}{x^2-1} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{3x^2+4x-4}{x^2-1} }\)

Solution

2961 video solution

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\(\displaystyle{ f(x) = \frac{x^2+2x-3}{x-2} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x^2+2x-3}{x-2} }\)

Solution

2962 video solution

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\(\displaystyle{ f(x) = \frac{2x}{x^2+3} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{2x}{x^2+3} }\)

Solution

2963 video solution

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\(\displaystyle{ f(x) = \frac{x-1}{x^2-x-12} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x-1}{x^2-x-12} }\)

Solution

He makes a really good point at the end of the video about how a graph can cross horizontal asymptotes but a graph will never cross a vertical asymptote and why.

2964 video solution

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\(\displaystyle{ f(x) = \frac{(x-6)(x+2)}{(x-6)(x+1)} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{(x-6)(x+2)}{(x-6)(x+1)} }\)

Solution

2965 video solution

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\(\displaystyle{ f(x) = \frac{(x-1)(x+3)}{x+2} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{(x-1)(x+3)}{x+2} }\)

Solution

2966 video solution

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\(\displaystyle{ f(x) = 3 + \frac{2}{x-1} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = 3 + \frac{2}{x-1} }\)

Solution

2967 video solution

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\(\displaystyle{ f(x) = \frac{2x+1}{x-2} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{2x+1}{x-2} }\)

Solution

2968 video solution

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\(\displaystyle{ f(x) = \frac{x-3}{2x+5} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x-3}{2x+5} }\)

Solution

2969 video solution

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\(\displaystyle{ y = \frac{-12}{x} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{-12}{x} }\)

Solution

2970 video solution

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\(\displaystyle{ y = \frac{3}{x+4} - 1 }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{3}{x+4} - 1 }\)

Solution

2971 video solution

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\(\displaystyle{ y = \frac{5}{x-2} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{5}{x-2} }\)

Solution

2972 video solution

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\(\displaystyle{ y = \frac{x+1}{x+2} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{x+1}{x+2} }\)

Solution

2973 video solution

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\(\displaystyle{ y = \frac{2x-3}{x-1} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{2x-3}{x-1} }\)

Solution

2974 video solution

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\(\displaystyle{ y = \frac{1}{x-3} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{1}{x-3} }\)

Solution

2975 video solution

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\(\displaystyle{ y = \frac{1}{x+2} + 7 }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{1}{x+2} + 7 }\)

Solution

2976 video solution

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\(\displaystyle{ y = \frac{6x-18}{2x+4} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{6x-18}{2x+4} }\)

Solution

2977 video solution

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\(\displaystyle{ y = \frac{2x^2-3x-2}{x^2+x-6} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{2x^2-3x-2}{x^2+x-6} }\)

Solution

2978 video solution

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\(\displaystyle{ y = \frac{2x^2-x+1}{x-2} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{2x^2-x+1}{x-2} }\)

Solution

2979 video solution

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\(\displaystyle{ y = \frac{x-1}{x^2+5x+6} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{x-1}{x^2+5x+6} }\).

Solution

2980 video solution

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\(\displaystyle{ y = \frac{x^2+3x-10}{x^2-4} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{x^2+3x-10}{x^2-4} }\)

Solution

2981 video solution

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\(\displaystyle{ y = \frac{4}{x^2+1} }\)

Problem Statement

Locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{4}{x^2+1} }\).

Solution

2982 video solution

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\(\displaystyle{f(x)=\frac{3}{2-x}}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{f(x)=\frac{3}{2-x}}\)

Solution

PatrickJMT - 1662 video solution

video by PatrickJMT

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\(\displaystyle{f(x)=\frac{4}{x}}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{f(x)=\frac{4}{x}}\)

Solution

MIP4U - 1663 video solution

video by MIP4U

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\(\displaystyle{f(x)=\frac{2x}{x-4}}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{f(x)=\frac{2x}{x-4}}\)

Solution

MIP4U - 1664 video solution

video by MIP4U

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\(\displaystyle{f(x)=\frac{2}{x+2}}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{f(x)=\frac{2}{x+2}}\)

Solution

MIP4U - 1665 video solution

video by MIP4U

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\(\displaystyle{f(x)=\frac{2}{(x+2)^2}}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{f(x)=\frac{2}{(x+2)^2}}\)

Solution

MIP4U - 1675 video solution

video by MIP4U

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\(\displaystyle{f(x)=\frac{x^2-9}{x-3}}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^2-9}{x-3}}\)

Solution

PatrickJMT - 1668 video solution

video by PatrickJMT

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\(\displaystyle{y=\frac{3x^2+4}{x-2}}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{y=\frac{3x^2+4}{x-2}}\)

Solution

PatrickJMT - 1670 video solution

video by PatrickJMT

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\(\displaystyle{f(x)=\frac{x+3}{x^2-9}}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x+3}{x^2-9}}\)

Solution

MIP4U - 1676 video solution

video by MIP4U

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\(\displaystyle{f(x)=\frac{8}{x^2-4}}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{f(x)=\frac{8}{x^2-4}}\)

Solution

MIP4U - 1680 video solution

video by MIP4U

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\(\displaystyle{y=\frac{(3x+12)(x-2)}{(x-1)(x+5)}}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{y=\frac{(3x+12)(x-2)}{(x-1)(x+5)}}\)

Solution

PatrickJMT - 1667 video solution

video by PatrickJMT

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\(\displaystyle{f(x)=\frac{x^3-6x^2+6x-36}{x^2-5x-6}}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^3-6x^2+6x-36}{x^2-5x-6}}\)

Solution

PatrickJMT - 1669 video solution

video by PatrickJMT

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\(\displaystyle{f(x)=\frac{1}{(x-2)}+1}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{f(x)=\frac{1}{(x-2)}+1}\)

Solution

MIP4U - 1678 video solution

video by MIP4U

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\(\displaystyle{f(x)=\frac{x^2-2x-3}{x^2-x-6}}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^2-2x-3}{x^2-x-6}}\)

Solution

MIP4U - 1679 video solution

video by MIP4U

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\(\displaystyle{y=\frac{x}{x^2-x-6}}\)

Problem Statement

Sketch a graph of the rational function \(\displaystyle{y=\frac{x}{x^2-x-6}}\)

Solution

Khan Academy - 1681 video solution

video by Khan Academy

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Practice Instructions

Sketch a graph of these rational functions by hand using the concepts of asymptotes, holes and zeroes discussed on this page.

Do NOT follow this link or you will be banned from the site!

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