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

17Calculus Precalculus - Graphing Rational Functions

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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) ≠ 0

n(x) = 0

d(x) ≠ 0

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.

Time for some practice problems.

Practice

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

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x^2}{x-2} }\).

Problem Statement

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x^2}{x-2} }\).

Solution

2948 video

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

Problem Statement

Unless otherwise instructed, 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

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

Problem Statement

Unless otherwise instructed, 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

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

Problem Statement

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

Solution

2962 video

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

Problem Statement

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

Solution

2963 video

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

Problem Statement

Unless otherwise instructed, 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

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

Problem Statement

Unless otherwise instructed, 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

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

Problem Statement

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

Solution

2966 video

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

Problem Statement

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

Solution

2967 video

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

Problem Statement

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

Solution

2968 video

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

Problem Statement

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

Solution

2969 video

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{-12}{x} }\).

Problem Statement

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{-12}{x} }\).

Solution

2970 video

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

Problem Statement

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

Solution

2971 video

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{5}{x-2} }\).

Problem Statement

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{5}{x-2} }\).

Solution

2972 video

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{x+1}{x+2} }\).

Problem Statement

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{x+1}{x+2} }\).

Solution

2973 video

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

Problem Statement

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

Solution

2974 video

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{1}{x-3} }\).

Problem Statement

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{1}{x-3} }\).

Solution

2975 video

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

Problem Statement

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

Solution

2976 video

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

Problem Statement

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

Solution

2977 video

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

Problem Statement

Unless otherwise instructed, 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

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

Problem Statement

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

Solution

2979 video

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

Problem Statement

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

Solution

2980 video

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

Problem Statement

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

Solution

2981 video

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

Problem Statement

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

Solution

2982 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{3}{2-x}}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{3}{2-x}}\) by hand using the concepts from the tutorial.

Solution

1662 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{4}{x}}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{4}{x}}\) by hand using the concepts from the tutorial.

Solution

1663 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{2x}{x-4}}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{2x}{x-4}}\) by hand using the concepts from the tutorial.

Solution

1664 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{2}{x+2}}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{2}{x+2}}\) by hand using the concepts from the tutorial.

Solution

1665 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{2}{(x+2)^2}}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{2}{(x+2)^2}}\) by hand using the concepts from the tutorial.

Solution

1675 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^2-9}{x-3}}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^2-9}{x-3}}\) by hand using the concepts from the tutorial.

Solution

1668 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{y=\frac{3x^2+4}{x-2}}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{y=\frac{3x^2+4}{x-2}}\) by hand using the concepts from the tutorial.

Solution

1670 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x+3}{x^2-9}}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x+3}{x^2-9}}\) by hand using the concepts from the tutorial.

Solution

1676 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{8}{x^2-4}}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{8}{x^2-4}}\) by hand using the concepts from the tutorial.

Solution

1680 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{y=\frac{(3x+12)(x-2)}{(x-1)(x+5)}}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{y=\frac{(3x+12)(x-2)}{(x-1)(x+5)}}\) by hand using the concepts from the tutorial.

Solution

1667 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^3-6x^2+6x-36}{x^2-5x-6}}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^3-6x^2+6x-36}{x^2-5x-6}}\) by hand using the concepts from the tutorial.

Solution

1669 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{1}{(x-2)}+1}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{1}{(x-2)}+1}\) by hand using the concepts from the tutorial.

Solution

1678 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^2-2x-3}{x^2-x-6}}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^2-2x-3}{x^2-x-6}}\) by hand using the concepts from the tutorial.

Solution

1679 video

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{y=\frac{x}{x^2-x-6}}\) by hand using the concepts from the tutorial.

Problem Statement

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{y=\frac{x}{x^2-x-6}}\) by hand using the concepts from the tutorial.

Solution

1681 video

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Really UNDERSTAND Precalculus

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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