Rational functions and asymptotes (vertical, horizontal, oblique)
Rational Functions and Asymptotes
Rational Functions and Asymptotes is a topic in Polynomial & Rational Functions in the Common Core State Standards. It is aligned to Standard F-IF.7d, which requires students to graph rational functions, identifying zeros and asymptotes.
A rational function is a ratio of polynomials \(\dfrac{p(x)}{q(x)}\); its graph has vertical asymptotes or holes from the denominator and a horizontal or slant asymptote from the degree comparison.
Theory
A rational function is a ratio of polynomials \(f(x)=\dfrac{p(x)}{q(x)}\). Its graph is shaped by where the denominator vanishes and by how the degrees compare.
- Vertical asymptote at each zero of \(q\) that does not cancel with \(p\).
- Hole at a zero of \(q\) that does cancel.
- Horizontal asymptote from degree comparison: top smaller \(\Rightarrow y=0\); equal \(\Rightarrow\) ratio of leading coefficients.
- Slant asymptote when \(\deg p=\deg q+1\); find it by division.
- Intercepts: \(y\)-intercept \(f(0)\); \(x\)-intercepts where \(p(x)=0\).
Asymptote rules for \(\dfrac{p(x)}{q(x)}\):
How to analyze a rational function
- Factor numerator and denominator.
- Cancel common factors — each gives a hole.
- Vertical asymptotes at the remaining denominator zeros.
- End behavior: compare degrees for a horizontal or slant asymptote.
- Intercepts, then sketch.
The denominator is zero at \(x=2\) (and does not cancel), giving a vertical asymptote. Numerator and denominator have equal degree, so the horizontal asymptote is the ratio of leading coefficients.
| \(\text{vertical}\) | \(:\) | \(x=2\) |
| \(\text{horizontal}\) | \(:\) | \(y=\dfrac{1}{1}=1\) |
Factor and cancel: the factor \((x-3)\) divides out.
| \(f(x)\) | \(=\) | \(\dfrac{(x-3)(x+3)}{x-3}\) |
| \(=\) | \(x+3,\quad x\neq 3\) |
So there is a hole at \((3,6)\), not a vertical asymptote.
Since the top degree is one more than the bottom, divide.
| \(\dfrac{x^2+1}{x}\) | \(=\) | \(x+\dfrac{1}{x}\) |
As \(x\to\pm\infty\), \(\dfrac{1}{x}\to 0\), so the graph approaches the line \(y=x\).
The \(y\)-intercept is \(f(0)\); the \(x\)-intercept makes the numerator zero.
| \(f(0)\) | \(=\) | \(\dfrac{-4}{2}=-2\) |
| \(x-4=0\) | \(\Rightarrow\) | \(x=4\) |
\(y\)-intercept \((0,-2)\); \(x\)-intercept \((4,0)\).
Common pitfalls
Frequently asked questions
How do you find vertical asymptotes?
Factor and cancel, then set the remaining denominator equal to zero. Each solution that did not cancel is a vertical asymptote.
How do you find the horizontal asymptote?
Compare degrees: smaller top gives \(y=0\); equal degrees give the ratio of leading coefficients; a larger top gives a slant asymptote instead.
What is a hole in a rational function?
A single missing point that occurs where a factor cancels from both numerator and denominator, such as \(\dfrac{x^2-9}{x-3}\) at \(x=3\).
What is a slant asymptote and when does it occur?
An oblique line the graph approaches, occurring when the numerator degree is exactly one more than the denominator degree. Find it by dividing.