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.

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

Solutions to the Above Exercises

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

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