## 17Calculus Precalculus - Graphing Rational 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) ≠ 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

video by PatrickJMT

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

video by MIP4U

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

video by MIP4U

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

video by MIP4U

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

video by MIP4U

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

video by PatrickJMT

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

video by PatrickJMT

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

video by MIP4U

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

video by MIP4U

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

video by PatrickJMT

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

video by PatrickJMT

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

video by MIP4U

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

video by MIP4U

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

video by Khan Academy

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