Graph Intersections Calculator

Calculator Inputs

Use degree 0 for constants, degree 1 for lines, and degree 2 for parabolas.

Graph A degree

Graph A quadratic coefficient a

Graph A linear coefficient b

Graph A constant c

Graph B degree

Graph B quadratic coefficient a

Graph B linear coefficient b

Graph B constant c

Decimal places

Graph A preview

Graph B preview

Example Data Table

Case	Graph A	Graph B	Expected intersections	Notes
Example 1	y = x² - 4	y = 2x	(-2, -4), (4, 8)	Two real crossings.
Example 2	y = x²	y = 2x - 1	(1, 1)	Tangent intersection.
Example 3	y = x² + 1	y = -x² - 3	No real point	Complex solutions only.

Formula Used

Let the two graphs be:

y₁ = a₁x² + b₁x + c₁

y₂ = a₂x² + b₂x + c₂

At an intersection, both y-values are equal:

a₁x² + b₁x + c₁ = a₂x² + b₂x + c₂

Move everything to one side:

(a₁ - a₂)x² + (b₁ - b₂)x + (c₁ - c₂) = 0

This creates a difference equation. Solve it using standard quadratic or linear rules.

If the difference is quadratic, use D = b² - 4ac.
If D > 0, there are two real intersections.
If D = 0, the graphs touch at one real point.
If D < 0, there is no real intersection on the graph.
Once x is known, substitute into either graph to find y.

How to Use This Calculator

Select the degree for Graph A and Graph B.
Enter the coefficients for each graph.
Use zero for unused higher-degree coefficients.
Choose the number of decimal places you want.
Click Calculate Intersections.
Read the classification, coordinates, and difference equation.
Download the result as CSV or PDF if needed.

FAQs

1. What graph types does this calculator support?

It supports constants, straight lines, and quadratic curves. Lower-degree graphs work by setting unused higher-order coefficients to zero automatically.

2. What does the discriminant tell me?

The discriminant shows how many real intersections exist after setting the equations equal. Positive means two, zero means one tangent point, and negative means no real meeting.

3. Why do I get no real intersection?

Your graphs may miss each other on the real plane. The solver may still show complex roots, which are algebraically valid but not visible as real graph crossings.

4. Can I enter decimals and negative numbers?

Yes. The calculator accepts decimal, positive, and negative coefficients, which is useful for scaled models, transformed parabolas, and shifted linear equations.

5. What happens if both graphs are identical?

The calculator reports infinite intersections. That means every point on one graph also lies on the other because both equations represent the same curve.

6. Why does the quadratic coefficient sometimes seem ignored?

If you choose degree 0 or degree 1, the calculator intentionally ignores higher-order terms. This keeps the selected graph type mathematically consistent.

7. Does graph order change the final intersection points?

No. Swapping Graph A and Graph B changes only the sign of the difference equation. The actual intersection coordinates remain the same.

8. Can I save the output for reports or homework?

Yes. Use the CSV button for spreadsheet-friendly data and the PDF button for printable summaries, revision sheets, or classroom submission records.