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 doublecheck 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 xaxis. To determine where the graph crosses the yaxis, 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 yaxis.
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 \(\displaystyle{ f(x) = \frac{x^2}{x2} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x^2}{x2} }\).
Solution 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x^2x6}{x^2 + x  2} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x^2x6}{x^2 + x  2} }\).
Solution 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{3x^2+4x4}{x^21} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{3x^2+4x4}{x^21} }\).
Solution 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x^2+2x3}{x2} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x^2+2x3}{x2} }\).
Solution 

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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 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x1}{x^2x12} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x1}{x^2x12} }\).
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.
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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{(x6)(x+2)}{(x6)(x+1)} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{(x6)(x+2)}{(x6)(x+1)} }\).
Solution 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{(x1)(x+3)}{x+2} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{(x1)(x+3)}{x+2} }\).
Solution 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = 3 + \frac{2}{x1} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = 3 + \frac{2}{x1} }\).
Solution 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{2x+1}{x2} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{2x+1}{x2} }\).
Solution 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x3}{2x+5} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ f(x) = \frac{x3}{2x+5} }\).
Solution 

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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 

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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 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{5}{x2} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{5}{x2} }\).
Solution 

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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 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{2x3}{x1} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{2x3}{x1} }\).
Solution 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{1}{x3} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{1}{x3} }\).
Solution 

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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 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{6x18}{2x+4} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{6x18}{2x+4} }\).
Solution 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{2x^23x2}{x^2+x6} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{2x^23x2}{x^2+x6} }\).
Solution 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{2x^2x+1}{x2} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{2x^2x+1}{x2} }\).
Solution 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{x1}{x^2+5x+6} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{x1}{x^2+5x+6} }\).
Solution 

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Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{x^2+3x10}{x^24} }\).
Problem Statement 

Unless otherwise instructed, locate all asymptotes, holes and zeroes and sketch a plot of \(\displaystyle{ y = \frac{x^2+3x10}{x^24} }\).
Solution 

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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 

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

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

video by PatrickJMT 

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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 

video by MIP4U 

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{2x}{x4}}\) by hand using the concepts from the tutorial.
Problem Statement 

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{2x}{x4}}\) by hand using the concepts from the tutorial.
Solution 

video by MIP4U 

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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 

video by MIP4U 

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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 

video by MIP4U 

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^29}{x3}}\) by hand using the concepts from the tutorial.
Problem Statement 

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^29}{x3}}\) by hand using the concepts from the tutorial.
Solution 

video by PatrickJMT 

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{y=\frac{3x^2+4}{x2}}\) by hand using the concepts from the tutorial.
Problem Statement 

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{y=\frac{3x^2+4}{x2}}\) by hand using the concepts from the tutorial.
Solution 

video by PatrickJMT 

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x+3}{x^29}}\) by hand using the concepts from the tutorial.
Problem Statement 

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x+3}{x^29}}\) by hand using the concepts from the tutorial.
Solution 

video by MIP4U 

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{8}{x^24}}\) by hand using the concepts from the tutorial.
Problem Statement 

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{8}{x^24}}\) by hand using the concepts from the tutorial.
Solution 

video by MIP4U 

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{y=\frac{(3x+12)(x2)}{(x1)(x+5)}}\) by hand using the concepts from the tutorial.
Problem Statement 

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{y=\frac{(3x+12)(x2)}{(x1)(x+5)}}\) by hand using the concepts from the tutorial.
Solution 

video by PatrickJMT 

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^36x^2+6x36}{x^25x6}}\) by hand using the concepts from the tutorial.
Problem Statement 

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^36x^2+6x36}{x^25x6}}\) by hand using the concepts from the tutorial.
Solution 

video by PatrickJMT 

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{1}{(x2)}+1}\) by hand using the concepts from the tutorial.
Problem Statement 

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{1}{(x2)}+1}\) by hand using the concepts from the tutorial.
Solution 

video by MIP4U 

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^22x3}{x^2x6}}\) by hand using the concepts from the tutorial.
Problem Statement 

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{f(x)=\frac{x^22x3}{x^2x6}}\) by hand using the concepts from the tutorial.
Solution 

video by MIP4U 

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Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{y=\frac{x}{x^2x6}}\) by hand using the concepts from the tutorial.
Problem Statement 

Unless otherwise instructed, sketch a graph of the rational function \(\displaystyle{y=\frac{x}{x^2x6}}\) by hand using the concepts from the tutorial.
Solution 

video by Khan Academy 

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Really UNDERSTAND Precalculus
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  doubleangle formulas  

\( \sin(2t) = 2\sin(t)\cos(t)\) 
\(\displaystyle{ \cos(2t) = \cos^2(t)  \sin^2(t) }\) 
Set 4  halfangle 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
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\) 
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\)  
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) 
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = \frac{1}{1+t^2} }\)  
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
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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