Graphing rational functions
Graphing Rational Functions
Graphing Rational Functions is a topic in Rational Functions in the California Common Core State Standards. It is aligned to Standard F-IF.7d, which requires students to graph rational functions, identifying zeros and asymptotes.
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.