High School Conic Section Identifier and Grapher
Analyze the general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0.
General Equation Coefficients
1x² + 1y² – 9 = 0
Conic Type
Discriminant (B² – 4AC)
Analysis
Conic Section Identifier and Grapher
Understanding the Conic Section Identifier and Grapher
Conic sections are the curves obtained by intersecting a cone with a plane. In algebra, they are described by the general quadratic equation in two variables: Ax² + Bxy + Cy² + Dx + Ey + F = 0. This tool helps you visualize these shapes instantly.
The Discriminant Test
To identify the shape without graphing, we calculate the discriminant, Δ = B² – 4AC.
- Ellipse: If Δ < 0. (A Circle is a special ellipse where A = C and B = 0).
- Parabola: If Δ = 0.
- Hyperbola: If Δ > 0.
Why use this Conic Section Identifier and Grapher?
Manual graphing of general conic equations (especially with an xy term, which indicates rotation) involves complex algebra like completing the square or coordinate rotation. This tool handles the heavy lifting, solving the quadratic for y at every point to render an accurate curve, allowing students to explore how changing coefficients transforms the shape.
FAQ
Yes, this tool is free and runs purely in the browser.
The B coefficient is attached to the xy term. A non-zero B rotates the conic section so its axes are not parallel to the x and y axes.
Sometimes the plane passes through the vertex of the cone, resulting in a point, a line, or two intersecting lines instead of a curve. This tool may show these as thin lines or dots.
Conics are infinite. The graph window is zoomed to -10 to 10. If the shape is outside this range, try adjusting D, E, or F to center it.