Calculator Inputs
Enter coefficients for two polynomial equations up to cubic form. The page keeps a working example ready, and the result appears above this form after submission.
Plotly Graph
The chart compares both equations across the selected x range and marks every real intersection point found by the calculator.
Example Data Table
This example uses the prefilled values: y₁ = x³ - 3x + 1 and y₂ = x + 1.
| Scenario | Equation 1 | Equation 2 | x Range | Intersections |
|---|---|---|---|---|
| Default example | y = x³ - 3x + 1 | y = x + 1 | -4 to 4 | (-2, -1), (0, 1), (2, 3) |
| Tangent example | y = x² | y = 2x - 1 | -3 to 3 | (1, 1) |
| No real crossing | y = x² + 4 | y = -x² - 5 | -5 to 5 | No real intersection |
Formula Used
Two graphs intersect where both y values are equal. The calculator sets f(x) = g(x) and moves every term to one side.
The working equation becomes h(x) = f(x) - g(x) = 0. The real roots of h(x) are the x coordinates of intersection points.
Each y coordinate is then found by substituting the root into either original equation. Residual error is measured with |f(x) - g(x)|.
How to Use This Calculator
- Enter coefficients for Equation 1 and Equation 2 in cubic form.
- Set the minimum and maximum x values for the graph window.
- Choose graph points and decimal precision.
- Press Calculate Intersections to show the result above the form.
- Review the table, graph, and intersection type labels.
- Download the results as CSV or PDF when needed.
FAQs
1. What kinds of equations does this calculator support?
It supports two polynomial equations up to cubic form. You can still model linear and quadratic equations by setting higher degree coefficients to zero.
2. What is an intersection point on a graph?
An intersection point is a coordinate where both equations produce the same y value for the same x value. That shared point lies on both graphs.
3. Why can two graphs have multiple intersections?
Higher degree equations can bend and change direction. That makes it possible for two curves to cross, touch, or separate several times across one viewing range.
4. What does “touching or tangent” mean?
A tangent intersection happens when both graphs meet at a point without crossing through each other. The difference equation has a repeated real root there.
5. Why do I see no real intersections?
Your equations may never share the same y value in the selected x range, or the real intersection may lie outside the range you entered.
6. What does residual mean in the results table?
Residual is the absolute difference between both y values at the computed intersection x value. Smaller residuals indicate a cleaner numerical match.
7. Can I use decimals and negative coefficients?
Yes. The inputs accept decimals, fractions converted to decimals, and negative values. This makes the tool useful for many classroom and applied math problems.
8. Why is the graph useful if I already have the table?
The graph shows how the curves behave between intersection points. It helps you verify crossings, spot tangencies, and understand the selected x range visually.