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# Graphs of Rational Functions

## Graphs of functions with x in the denominator of a fraction

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Graphs of Rational Functions

Can you use your graphing calculator to help you sketch the graph of the rational function? Does this function have any zeros or asymptotes?

y=4x2+1\begin{align*}y=\frac{4}{x^2+1}\end{align*}

### Graphing Rational Functions

A rational function is the quotient of two polynomial functions. In general,

f(x)=A(x)B(x)\begin{align*}f(x)=\frac{A(x)}{B(x)}\end{align*}

where A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are polynomials and B0\begin{align*}B \ne 0\end{align*}. You can use your graphing calculator to graph a rational function and look for important features. Consider the function y=xx3\begin{align*}y=\frac{x}{x-3}\end{align*}.

Notice that the function has two pieces. In between those two pieces are the asymptotes.

• A vertical asymptote occurs at the x\begin{align*}x\end{align*} value(s) that cause the denominator of the function to be equal to zero (which is undefined). This function has a vertical asymptote at x=3\begin{align*}x=3\end{align*}.
• A horizontal asymptote occurs at the y\begin{align*}y\end{align*} value(s) that cause the denominator of the function to be zero if the function is rewritten and solved for x\begin{align*}x\end{align*} instead of y\begin{align*}y\end{align*}. This is what it looks like to solve for x\begin{align*}x\end{align*}:
y=xx3(x3)(y)=(x3)(xx3)xy3y=(x3)(xx3)xy3y=xxyx=3yx(y1)=3yx=3yy1\begin{align*}& y=\frac{x}{x-3}\\ &(x-3)(y)=(x-3)\left(\frac{x}{x-3}\right)\\ & xy-3y=\cancel{(x-3)}\left(\frac{x}{\cancel{x-3}}\right)\\ & xy-3y=x\\ & xy-x=3y\\ & x(y-1)=3y\\ &\boxed{x=\frac{3y}{y-1}}\end{align*}
Thus, y1\begin{align*}y \ne 1\end{align*} and this function has a horizontal asymptote at y=1\begin{align*}y = 1\end{align*}.

The image below shows the graph with the asymptotes drawn in and labelled. For rational functions, the asymptotes represent the lines that the function will approach but never touch.

It is also important to note the x-intercepts (zeros) of the function. The zeros of the function will be the values for x\begin{align*}x\end{align*} that cause the numerator, but not also the denominator, to be equal to zero.

#### Let's graph the following rational functions:

1. y=1x29=1(x+3)(x3)\begin{align*}y=\frac{1}{x^2-9}=\frac{1}{(x+3)(x-3)}\end{align*}

Here is the sketch from the calculator:

It can sometimes be hard to interpret what you see on the graphing calculator screen. Use algebra to find the asymptotes and zeros and sketch those first.

There are no zeros for the numerator. Therefore, there are no x-intercepts for the function.

The zeros of the denominator are 3 and –3. This means that there are two vertical asymptotes. One vertical asymptote is the line x=3\begin{align*}x = 3\end{align*} and the other is the line x=3\begin{align*}x = -3\end{align*}.

Another way to determine a horizontal asymptote besides solving the equation for x\begin{align*}x\end{align*} is to look at the degrees of the numerators and denominators. The degree is the highest exponent. The degree of the numerator is 0 and the degree of the denominator is 2. In general, if the degree of the numerator is less than the degree of the denominator, there will be a horizontal asymptote at y=0\begin{align*}y=0\end{align*}.

Once you have sketched the asymptotes, use the table and/or graph from the graphing calculator to decide what the rest of the graph looks like. Here is the graph with the asymptotes labelled.

1. y=1x\begin{align*}y=-\frac{1}{x}\end{align*}

Here is the sketch from the calculator:

Use algebra to find the asymptotes and zeros and sketch those first.

There are no zeros for the numerator. Therefore, there are no x-intercepts for the function.

The zero of the denominator is 0. This means that there is one vertical asymptote, the line x=0\begin{align*}x = 0\end{align*}.

The horizontal asymptote is y=0\begin{align*}y = 0\end{align*}. You can use algebra to solve the equation for x\begin{align*}x\end{align*} and look for the values of y\begin{align*}y\end{align*} that will cause the denominator to be equal to zero:

y=1xxy=1xyy=1yx=1y\begin{align*}& y=-\frac{1}{x}\\ & xy=-1\\ &\frac{xy}{y}=\frac{-1}{y}\\ &\boxed{x=-\frac{1}{y}}\end{align*}

Once you have sketched the asymptotes, use the table and/or graph from the graphing calculator to decide what the rest of the graph looks like. Here is the graph with the asymptotes labelled.

1. y=x+2x3\begin{align*}y=\frac{x+2}{x-3}\end{align*}

Here is the sketch from the calculator:

Use algebra to find the asymptotes and zeros and sketch those first.

The zero of the numerator is –2 so the zero (x-intercept) of the function is (–2, 0).

The zero of the denominator is 3. This means that there is one vertical asymptote, the line x=3\begin{align*}x = 3\end{align*}.

The horizontal asymptote is y=1\begin{align*}y = 1\end{align*}. You can use algebra to solve the equation for x\begin{align*}x\end{align*} and look for the values of y\begin{align*}y\end{align*} that will cause the denominator to be equal to zero:

\begin{align*}& y=\frac{x+2}{x-3}\\ &(x-3)(y)=\left(\frac{x+2}{x-3}\right)(x-3)\\ & xy-3y=x+2\\ & xy-x=2+3y\\ & x(y-1)=2+3y\\ &\boxed{x=\frac{3y+2}{y-1}}\end{align*}

Once you have sketched the asymptotes, use the table and/or graph from the graphing calculator to decide what the rest of the graph looks like. Here is the graph with the asymptotes labelled.

### Examples

#### Example 1

Earlier, you were asked if the following function has any zeros or asymptotes:

\begin{align*}y=\frac{4}{x^2+1}\end{align*}

Here is a sketch of the function:

• There are no zeros for this function since there are no zeros for the numerator. The graph does not cross the \begin{align*}x\end{align*}-axis.
• There are no zeros for the denominator. Therefore, there are no vertical asymptotes.
• Because the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at \begin{align*}y=0\end{align*}.

#### Example 2

Sketch the graph of the rational function: \begin{align*}y=\frac{x+1}{x}\end{align*}

The zero of the denominator is 0. This means that there will be a vertical asymptote at \begin{align*}x=0\end{align*} (the y-axis). Because the degree of the denominator and numerator are the same, you can solve the equation for \begin{align*}x\end{align*} and get \begin{align*}x=\frac{1}{y-1}\end{align*}. The zero of the denominator is now 1. This means there is a horizontal asymptote at \begin{align*}y=1\end{align*}. The numerator has a zero at -1, so there is an x-intercept at -1.

#### Example 3

Sketch the graph of the rational function: \begin{align*}y=\frac{x^2}{x-3}\end{align*}

The zero of the denominator is 3. This means that there will be a vertical asymptote at \begin{align*}x=3\end{align*}. Because the degree of the numerator is greater than the degree of the denominator, there are no horizontal asymptotes. The zero of the numerator is 0, so there is an x-intercept at 0.

#### Example 4

Find the asymptotes of the function: \begin{align*}y=\frac{1}{x^2-16}\end{align*}

The zeros of the denominator are 4 and –4. This means that there will be two vertical asymptotes. One vertical asymptote will be the line \begin{align*}x = 4\end{align*} and the other will be the line \begin{align*}x = -4\end{align*}. Because the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at \begin{align*}y=0\end{align*}. Though you did not need to sketch the graph, here is the graph of the function:

### Review

Sketch the graph of each of the following rational functions.

1. \begin{align*}y=\frac{2}{x+3}\end{align*}
2. \begin{align*}y=\frac{x}{x-1}\end{align*}
3. \begin{align*}y=\frac{1}{x^2-4}\end{align*}
4. \begin{align*}y=\frac{x+2}{x}\end{align*}
5. \begin{align*}y=\frac{1}{x^2+2}\end{align*}
6. \begin{align*}y=\frac{x}{x+2}\end{align*}
7. \begin{align*}y=\frac{1}{x^2-x-12}\end{align*}
8. \begin{align*}y=\frac{x-1}{x+3}\end{align*}
9. \begin{align*}y=\frac{x-1}{x+4}\end{align*}
10. \begin{align*}y=\frac{5}{x^2+1}\end{align*}

Without graphing the following rational functions, state what you know about their asymptotes and zeros.

1. \begin{align*}y=\frac{1}{x^2-x-2}\end{align*}
2. \begin{align*}y=-\frac{2}{x-4}\end{align*}
3. \begin{align*}y=-\frac{2}{x^2+1}\end{align*}
4. \begin{align*}y=\frac{6}{x^2+1}\end{align*}
5. \begin{align*}y=\frac{x-1}{x+3}\end{align*}

To see the Review answers, open this PDF file and look for section 8.4.

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### Vocabulary Language: English

compression

A stretch or compression is a function transformation that makes a graph narrower or wider, without translating it horizontally or vertically.

Function

A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that indicates where a function flattens out as the independent variable gets very large or very small. A function may touch or pass through a horizontal asymptote.

Oblique Asymptote

An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. An oblique asymptote may be found through long division.

Polynomial Function

A polynomial function is a function defined by an expression with at least one algebraic term.

Reflection

Reflections are transformations that result in a "mirror image" of a parent function. They are caused by differing signs between parent and child functions.

shift

A shift, also known as a translation or a slide, is a transformation applied to the graph of a function that does not change the shape or orientation of the graph, only the location of the graph.

shifts

A shift, also known as a translation or a slide, is a transformation applied to the graph of a function that does not change the shape or orientation of the graph, only the location of the graph.

Slant Asymptote

A slant asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but will never reach. A slant asymptote exists when the numerator of the function is exactly one degree greater than the denominator. A slant asymptote may be found through long division.

stretch

A stretch or compression is a function transformation that makes a graph narrower or wider.

Transformations

Transformations are used to change the graph of a parent function into the graph of a more complex function.

Vertical Asymptote

A vertical asymptote is a vertical line marking a specific value toward which the graph of a function may approach, but will never reach.

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