Graphing rational functions
Graphing Rational Functions
Graphing Rational Functions is a topic in Rational Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.6(G), which requires students to analyze the effect on the graph of changing parameters of a rational function.
Graphing a rational function uses its vertical and horizontal asymptotes, its intercepts, and a test point in each region.
Theory
To graph a rational function, assemble its key features:
- Vertical asymptotes: uncancelled denominator zeros.
- Horizontal asymptote: from comparing degrees.
- Intercepts: numerator zero for \(x\)-intercept, \(f(0)\) for \(y\)-intercept.
- Test points in each region between asymptotes.
Intercepts:
How to graph
- Find vertical asymptotes.
- Find the horizontal (or slant) asymptote.
- Find the intercepts.
- Plot test points and sketch each branch.
VA where denominator is zero; HA from equal degrees.
| \(\text{VA}\) | \(:\) | \(x=2\) |
| \(\text{HA}\) | \(:\) | \(y=1\) |
| \(\text{x-int}\) | \(:\) | \(x=-1\) |
| \(\text{y-int}\) | \(:\) | \(y=-\dfrac12\) |
Equal degrees give \(y=3\).
| \(\text{VA}\) | \(:\) | \(x=-4\) |
| \(\text{HA}\) | \(:\) | \(y=3\) |
Shift the parent right \(1\), up \(2\).
| \(\text{VA}\) | \(:\) | \(x=1\) |
| \(\text{HA}\) | \(:\) | \(y=2\) |
The graph crosses where the numerator is zero.
| \(x-3\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(3\) |
Common pitfalls
Frequently asked questions
How do you graph a rational function?
Find the asymptotes and intercepts, then plot test points.
How do you find the x-intercept?
Set the numerator equal to zero.
Can a graph cross a horizontal asymptote?
Yes, in the middle — it only can't at the far ends.
What guides the sketch?
The asymptotes and a test point in each region.