# Slant Asymptotes of Rational Functions - Interactive

An online graphing calculator to graph
**rational functions** of the form \( f(x) = \dfrac{a x^2 + b x + c}{d x + e} \) by entering different values for the parameters \( a , b , c , d \) and \( e \). When the degree of the numerator is one unit higher that the degree of the denominator, as is the case with function \( f \) defined above, the graph of the rational function has a **slant asymptote** which a line.

Example

Find the slant asymptote of the rational function givern by \( f(x) = \dfrac{3 x^2 + 2 x - 1}{ -2 x + 2} \)

Being a ratio of two polynomials and using the division of polynomials, the given function mwy be written
\[ f(x) = Q(x) + \dfrac{R}{-2x+2} \]
where \( Q(x) \) is the quotient and \( R \) is the remainder of the division of the numerator by the denominator.

For the given function and using polynomials division, we have

\[ f(x) = Q(x) + \dfrac{R}{-2x+2} = -\dfrac{3x}{2}-\dfrac{5}{2}+\dfrac{4}{-2x+2} \]

As \( x \) increases indefinitely (\( + \infty\) ) or decreases indefinitely (\( - \infty\) ), the term \( \dfrac{4}{-2x+2} \) approaches zero and the graph of \( f(x) \) will be close to the graph of the line \( y = Q(x) = -\dfrac{3x}{2}-\dfrac{5}{2} \) which is called the **slant asymptote**.

Note that in order to have graphs with slant asymptotes, both parameters \( a \) and \( d \) must be different from zero.

## Use The Graphing Calculator to Graph Rational Functions and Explore the Slant Asymptote

Enter values for the parameters \( a , b , c , d \) and \( e \) and press on "Graph". The graph of \( f \) , the slant and vertical asymptotes will be displayed.

## Interactive Tutorials

Let the slant asymptote have the equation: \( y = M x + B \), where \( M \) is the slope and \( B \) is the y -intercept.

a) Use different values of \( a , b , c , d \) and \( e \) and show that the slope \( M \) depends on the parameters \( a\) and \( d \) only.

b) Use different values of \( a , b , c , d \) and \( e \) and show that \( B \) does not depend on the parameter \( c \).

c) Using polynomial division, it can be shown that:

\( M = \dfrac{a}{d} \) and \( B = \dfrac{1}{d}(b - \dfrac{e a}{d}) \)

Use different values of \( a , b , c , d \) and \( e \); approximate \( M \) and \( B \) from the graph given by the graphing calculator and use the formulas above to verify that the values of \( M \) and \( B \) given by the graph are close to those given by the formulas.

## Exercises

For each function, find the slant asymptote analytically and check your answer graphically using the graphing calculator above.

- \( f(x) = \dfrac{ - x^2 + 2 x - 1}{ - x + 2} \)

- \( g(x) = \dfrac{ 2 x^2 + 3 x - 1}{ - 4 x + 2} \)

- \( h(x) = \dfrac{ x^2 + 2x + 3 }{ 2x + 2} \)

## Solutions to the Above Exercises

- \( y = x \)

- \( y = -\dfrac{x}{2}-1 \)

- \( y = \dfrac{x}{2}+\dfrac{1}{2} \)

## More References and Links

rational functionsVertical Asymptotes of Rational Functions - Interactive

Horizontal Asymptotes of Rational Functions - Interactive

Graphing Calculators .