Quadratic Equations Lab: Coefficients, Roots, and Vertex
Change quadratic coefficients to compare real roots, the discriminant, the vertex, and the graph.
Free, with no sign-up.
For High school students and adults reviewing the topic.
About 10 minutes.
How does it work?
- Change the controls or open a named preset. Read the outputs, then check a challenge.
What will you get?
Compare the model’s outputs and check two challenges.
What does the model assume?
- The function is y = ax² + bx + c with real coefficients and a nonzero a. Setting a to zero does not define a quadratic and produces no result.
- Outputs describe the mathematical function, not a measured physical system. One repeated real root is counted as one distinct root.
- The vertex lies at x = −b/(2a). The discriminant b² − 4ac determines whether there are two, one, or no real roots.
- The graph scales to include the vertex and real roots. Squares mark roots and a circle marks the vertex; overlapping squares represent a repeated root.
Which setup would you like to try?
What can you change?
Range: -5 to 5 .
Range: -20 to 20 .
Range: -30 to 30 .
What does the model predict?
| Quantity | Value | Unit |
|---|---|---|
| Discriminant | 36.0000 | |
| Distinct real roots | 2.0000 | |
| Lower real root | -1.0000 | |
| Upper real root | 5.0000 | |
| Vertex x-coordinate | 2.0000 | |
| Vertex y-coordinate | -9.0000 | |
| Axis of symmetry x-coordinate | 2.0000 |
Parabola with 2 distinct real roots and vertex (2, -9).
Solid curve: function. Dashed line: axis of symmetry. Squares: roots. Circle: vertex.
Can you reach these targets?
Keep a = 1 and b = 6. Choose c so the graph just touches the x-axis.
- Target: Squared-term coefficient a = 1 ; tolerance 0 .
- Target: Linear coefficient b = 6 ; tolerance 0 .
- Target: Discriminant = 0 ; tolerance 0 .
Set a = −1 and b = 4. Choose c so the maximum height is 7.
- Target: Squared-term coefficient a = -1 ; tolerance 0 .
- Target: Linear coefficient b = 4 ; tolerance 0 .
- Target: Vertex y-coordinate = 7 ; tolerance 0.0001 .