Rational Graph Feature Form
Example Data Table
| Numerator | Denominator | Main features |
|---|
| 1, 0, -1 | 1, -3, 2 | Hole at x = 1, vertical asymptote at x = 2. |
| 2, 3 | 1, -4 | Vertical asymptote at x = 4. Horizontal asymptote y = 2. |
| 1, 0, 0 | 1, -1 | Slant asymptote appears after division. |
| 1, -5, 6 | 1, -1 | Zeros at x = 2 and x = 3 in the scan window. |
Formula Used
A rational function has the form f(x) = P(x) / Q(x), where Q(x) is not zero.
Domain restrictions come from Q(x) = 0. Zeros come from P(x) = 0 after cancelled factors are checked.
Vertical asymptotes come from the remaining denominator roots. Holes come from common factors removed from P(x) and Q(x).
Horizontal asymptotes use degrees. If deg P is less than deg Q, y = 0. If degrees match, y equals the leading coefficient ratio.
When deg P is greater than deg Q, polynomial division gives the slant or polynomial asymptote. Critical points use f'(x) = (P'Q - PQ') / Q².
How to Use This Calculator
Write each polynomial as descending coefficients. Use commas, spaces, or line breaks. Put missing powers as zero.
Choose an x-window wide enough to include expected roots. Increase samples for harder graphs or close roots.
Set a smaller tolerance when coefficients are exact and roots are close. Use the y-limit to make the graph readable.
Press the submit button. The result appears above the form. Use the export buttons to save the summary.
Understanding Rational Graph Features
A rational graph is built from a fraction of polynomials. The numerator controls many zeros. The denominator controls restrictions. This tool studies both parts together. It first reads the coefficient lists. Then it forms the polynomial functions. The order of coefficients matters. Always start with the highest power. Add zero when a power is missing.
Domain and Discontinuities
The domain excludes values that make the denominator zero. These values may create vertical asymptotes or holes. A vertical asymptote remains after common factors are removed. A hole appears when the same factor exists in both polynomials. The graph still cannot include that x-value. The missing point has a y-value from the simplified expression.
Intercepts and Zeros
An x-intercept happens when the simplified numerator equals zero. It must also be allowed by the original domain. This rule prevents a removed point from being counted as an intercept. The y-intercept is found by using x = 0. It exists only when the denominator is not zero there. These intercepts give useful anchor points.
Asymptotes and End Behavior
Asymptotes describe the shape when the graph moves near restrictions or far from the origin. Vertical asymptotes come from remaining denominator zeros. Horizontal asymptotes use degree comparison. If the numerator degree is smaller, the graph approaches y = 0. If the degrees match, it approaches the ratio of leading coefficients. If the numerator degree is larger, division gives a slant or polynomial asymptote.
Advanced Checks
The calculator also estimates critical points. It uses the derivative numerator P'Q - PQ'. Roots of this expression mark possible turning points. The sign chart tests intervals around zeros and asymptotes. These checks help you sketch the graph with fewer mistakes. Numerical scanning can miss roots outside the selected window. A wider range and more samples improve detection. Exact algebra is still useful for final proof.
Practical Workflow
Begin with simple features before drawing the curve. Mark domain restrictions first. Add intercepts next. Place holes and vertical asymptotes carefully. Then compare degrees for end behavior. Use the sample table to verify points on each interval. Finally, read the sign chart. This order creates a cleaner sketch and reduces algebra errors during study or review sessions each time.
FAQs
What coefficients should I enter?
Enter coefficients from highest power to constant term. For x² - 1, enter 1, 0, -1. The zero keeps the missing x term in place.
Can this find holes?
Yes. It searches for common real factors in the numerator and denominator. A removed common factor can create a hole.
Why is a zero not an x-intercept?
A zero is not an intercept when the original denominator is also zero there. That point is outside the domain.
How are vertical asymptotes found?
The calculator cancels common factors first. Then it finds real roots of the remaining denominator inside the selected window.
How are horizontal asymptotes found?
They are based on degrees. A smaller numerator gives y = 0. Equal degrees give the leading coefficient ratio.
What if my graph has a slant asymptote?
When the numerator degree is one more than the denominator degree, polynomial division gives the slant line.
Why should I change root scan samples?
More samples help detect close roots and sharp changes. Use higher values for complex polynomials or narrow features.
Can I export my results?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a printable report.