Slant Asymptotes of Rational Functions - Interactive
An online graphing calculator to graph rational functions of the form:
\[f(x) = \dfrac{a x^2 + b x + c}{d x + e}\]When the degree of the numerator is exactly one unit higher than the degree of the denominator (as is the case with function \(f\) defined above when \(a \ne 0\) and \(d \ne 0\)), the graph of the rational function has a slant (or oblique) asymptote, which is a slanted line.
Example Walkthrough
Find the slant asymptote of the rational function given by:
\[f(x) = \dfrac{3 x^2 + 2 x - 1}{ -2 x + 2}\]Being a ratio of two polynomials, the given function may be written using polynomial long division as:
\[f(x) = Q(x) + \dfrac{R}{-2x+2}\]where \(Q(x)\) is the quotient and \(R\) is the remainder. For the given function, polynomial division yields:
\[f(x) = -\dfrac{3x}{2} - \dfrac{5}{2} + \dfrac{4}{-2x+2}\]As \(x\) increases indefinitely (\(+ \infty\)) or decreases indefinitely (\(- \infty\)), the remainder term \(\dfrac{4}{-2x+2}\) approaches zero. Thus, the graph of \(f(x)\) will get closer and closer to the graph of the line:
\[y = Q(x) = -\dfrac{3x}{2} - \dfrac{5}{2}\]This line is called the slant asymptote.
Interactive Graphing Calculator
Enter values for the parameters \(a , b , c , d\) and \(e\) and click "Graph". The graph of \(f\), the slant asymptote, and the vertical asymptote will be displayed dynamically.
Hover your mouse cursor over the graph to trace coordinates. Use the toolbar to zoom, pan, or download as a PNG.
Interactive Tutorials
Let the slant asymptote have the equation: \(y = M x + B\), where \(M\) is the slope and \(B\) is the y-intercept.
- Use different values in the calculator above and observe that the slope \(M\) depends only on parameters \(a\) and \(d\).
- Observe that the y-intercept \(B\) does not depend on the parameter \(c\).
- Using polynomial division, it can be proven that the slope and y-intercept are given by the formulas: \[M = \dfrac{a}{d} \quad \text{and} \quad B = \dfrac{1}{d}\left(b - \dfrac{e a}{d}\right)\]
- Test this out: Approximate \(M\) and \(B\) from the graph visually, then use the formulas above to verify they match perfectly.
Practice Exercises
For each function, find the slant asymptote analytically using polynomial division. Then, check your answers graphically using the graphing calculator above.
- \(f(x) = \dfrac{ - x^2 + 2 x - 1}{ - x + 2}\)
- \(g(x) = \dfrac{ 2 x^2 + 3 x - 1}{ - 4 x + 2}\)
- \(h(x) = \dfrac{ x^2 + 2x + 3 }{ 2x + 2}\)
View Solutions
- Dividing \(-x^2 + 2x - 1\) by \(-x + 2\) yields a quotient of \(x\).
Slant Asymptote: \(y = x\) - Dividing \(2x^2 + 3x - 1\) by \(-4x + 2\) yields a quotient of \(-\dfrac{1}{2}x - 1\).
Slant Asymptote: \(y = -\dfrac{x}{2} - 1\) - Dividing \(x^2 + 2x + 3\) by \(2x + 2\) yields a quotient of \(\dfrac{1}{2}x + \dfrac{1}{2}\).
Slant Asymptote: \(y = \dfrac{x}{2} + \dfrac{1}{2}\)