Sign of a quadratic function
MathematicsAlgebraAges 15–16
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Algebra
Drag a, b and c to change the parabola y = ax² + bx + c and read the matching sign table right away. The simulation shades the part of the graph above the x-axis in green and the part below in red, so it becomes clear why the sign of f(x) depends only on the sign of a and on how the discriminant Δ compares with 0.
Lesson: Sign of a quadratic function
What it shows
The sign of a quadratic function f(x) = ax² + bx + c can be read from its graph: f(x) > 0 where the parabola lies above the x-axis and f(x) < 0 where it lies below. The discriminant Δ = b² − 4ac decides how many roots there are. If Δ < 0, f(x) has the sign of a for every x; if Δ = 0, it has the sign of a everywhere except at the double root; if Δ > 0, it has the opposite sign of a between the two roots and the sign of a outside them. This rule is the basis for solving quadratic inequalities.
How to use
Drag Coefficient a, Coefficient b and Coefficient c and watch the green (positive) and red (negative) parts of the parabola and the sign table update together. Change the sign of a to see the table reverse. Click Adjust c so Δ = 0 to show the case where the graph just touches the axis. For practice, switch off Show sign table, let students build the table from the graph, then switch it back on to check.
Parameters you can change
- Coefficient a -4–4
- Coefficient b -8–8
- Coefficient c -8–8
- Show sign table
Questions to explore
- How does the sign table change when you change the sign of a but keep b and c?
- For f(x) = x² − x + c, which values of c make f(x) positive for every x?
- When Δ < 0, why can f(x) never take the opposite sign of a?
