Calculator Form
Example Data Table
| Numerator coefficients | Denominator coefficients | Function idea | Hole coordinate |
|---|---|---|---|
| 1,-5,6 | 1,-1,-6 | (x - 2)(x - 3) / ((x - 3)(x + 2)) | (3, 0.2) |
| 1,0,-1 | 1,-1 | (x - 1)(x + 1) / (x - 1) | (1, 2) |
| 1,-4,4 | 1,-2 | (x - 2)^2 / (x - 2) | (2, 0) |
Formula Used
A hole appears when a rational function has a common factor in the numerator and denominator.
Let f(x) = N(x) / D(x). If N(a) = 0 and D(a) = 0, then x = a may create a hole.
Cancel the shared factor. Then evaluate the simplified function at x = a.
Hole coordinate = (a, simplified N(a) / simplified D(a)).
If the simplified denominator is still zero, the point is not a removable hole.
How to Use This Calculator
Enter numerator coefficients from highest degree to constant term.
Enter denominator coefficients in the same order.
Add optional suspected x-values when roots are decimal or known.
Set decimal places and tolerance if needed.
Press Submit to show the result above the form.
Use CSV or PDF buttons to save the result.
About Finding Coordinates of a Hole
What a Hole Means
A hole is a missing point on a rational function. It happens when the numerator and denominator share a factor. The graph looks almost continuous there. Yet the original function is not defined at that x-value. The missing point has an x-coordinate and a y-coordinate.
Why Common Factors Matter
The shared factor creates zero in both parts of the fraction. This makes the original expression undefined. After cancellation, the remaining expression shows the curve that the graph follows. The hole sits where the canceled factor equals zero.
How the Calculator Works
This calculator reads polynomial coefficients. It searches rational roots from the numerator and denominator. It also checks any manual x-values you enter. When a root appears in both polynomials, the tool cancels the common factor. Then it evaluates the simplified expression at that x-value.
Understanding the Result
The x-coordinate comes from the canceled factor. The y-coordinate comes from the simplified function. For example, if the canceled factor is x - 3, the x-coordinate is 3. If the simplified function gives 0.2 at x = 3, the hole is (3, 0.2).
When a Hole Is Not Removable
Some shared factors do not remove every denominator zero. If a denominator factor remains after cancellation, the graph has a vertical issue instead. The calculator marks that case as not removable. This helps avoid a common algebra mistake.
Useful Study Notes
Always factor first when possible. Then cancel only matching factors, not separate terms. A hole is tied to the original domain restriction. The simplified equation helps find the y-value. Both pieces are needed for a complete answer.
Exporting Work
The CSV export is useful for spreadsheets. The PDF export is useful for notes and assignments. The output includes the original function, simplified function, coordinates, and status notes.
FAQs
What is a hole in a rational function?
A hole is a missing point caused by a factor that cancels from the numerator and denominator. The original function is undefined there, but the simplified function gives the y-coordinate.
How do I find the x-coordinate of a hole?
Find the common factor. Set that factor equal to zero. Solve for x. That value is the x-coordinate of the hole.
How do I find the y-coordinate?
Cancel the shared factor first. Substitute the x-coordinate into the simplified rational function. The result is the y-coordinate.
Can there be more than one hole?
Yes. A rational function can have multiple holes when it has multiple shared factors. Each canceled factor can create its own missing point.
What coefficient order should I enter?
Enter coefficients from highest degree to constant term. For x² - 5x + 6, enter 1,-5,6.
Why should I use manual x-values?
Manual values help when roots are decimals, irrational approximations, or already known from another method. The calculator tests those values directly.
Is a vertical asymptote the same as a hole?
No. A hole is removable after cancellation. A vertical asymptote remains when the simplified denominator is still zero at that x-value.
Can I export my result?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a simple printable record of the calculation.