Calculator Input
Formula Used
A rational function is written as f(x) = N(x) / D(x), where N(x) is the numerator polynomial and D(x) is the denominator polynomial.
X-intercepts: N(x) = 0 after cancelling common factors
Vertical asymptotes: D(x) = 0 after cancelling common factors
Holes: common factor roots removed from both N(x) and D(x)
Horizontal asymptote:
• degree(N) < degree(D) → y = 0
• degree(N) = degree(D) → y = leading(N) / leading(D)
Slant or polynomial asymptote: quotient from N(x) ÷ D(x)
Polynomial division is also used to find end behavior when the numerator degree is at least the denominator degree.
How to Use This Calculator
- Enter numerator coefficients from x³ down to the constant term.
- Enter denominator coefficients from x³ down to the constant term.
- Set the graph window using minimum and maximum x-values.
- Choose plot point density and decimal precision.
- Press Graph Function to show the result above the form.
- Review intercepts, holes, asymptotes, simplified form, and the generated Plotly graph.
- Download the sample table as CSV or export the report as PDF.
Example Data Table
Example function: f(x) = (x² - 1) / (x - 1). This simplifies to x + 1, but it has a hole at x = 1.
| X | N(x) = x² - 1 | D(x) = x - 1 | f(x) | Observation |
|---|---|---|---|---|
| 0 | -1 | -1 | 1 | Valid point |
| 0.5 | -0.75 | -0.5 | 1.5 | Valid point |
| 1 | 0 | 0 | Undefined | Hole location |
| 2 | 3 | 1 | 3 | Valid point |
| 3 | 8 | 2 | 4 | Valid point |
Frequently Asked Questions
1) What does this graphing rational functions calculator do?
It graphs a rational function, simplifies common factors, estimates holes, finds intercepts, checks domain restrictions, and marks asymptote behavior from the entered coefficients.
2) How do I enter the function?
Enter the numerator and denominator coefficients from x³ to the constant term. For missing powers, simply enter zero in that coefficient field.
3) How to find the holes of a rational function calculator?
Holes occur when numerator and denominator share the same factor. After cancelling that factor, the excluded x-value becomes a hole, and the simplified function gives its y-coordinate.
4) How are vertical asymptotes found?
Vertical asymptotes come from denominator roots that remain after cancelling shared factors. If a denominator zero is not removed, the graph approaches positive or negative infinity there.
5) How are x-intercepts different from holes?
X-intercepts come from simplified numerator zeros that still belong to the graph. Holes are removed points caused by common factors, so they are not true intercepts.
6) Can this calculator show horizontal or slant asymptotes?
Yes. It compares numerator and denominator degrees. Equal degrees give a horizontal asymptote, one degree higher gives a slant asymptote, and larger differences give polynomial end behavior.
7) Why are some table values undefined?
A value is undefined whenever the denominator becomes zero at that x-value. That usually indicates a hole or a vertical asymptote location.
8) What export options are included?
You can download the generated sample data table as CSV and export the report area as PDF. A print-friendly report is also included.