Quadratic function y = ax² + bx + c: vertex, axis of symmetry, variation table
MathematicsFunctions & GraphsAges 15–16
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Functions & Graphs
The parabola y = ax² + bx + c is graphed with three sliders for a, b, c right in the simulation; students can also drag the vertex I directly to change b and c. The vertex I(−b/2a; −Δ/4a), the axis of symmetry x = −b/2a, the intercepts with both axes, Δ, and the variation table (arrows pointing up or down according to the sign of a) are drawn automatically and update instantly.
Lesson: Quadratic functions and their graphs
What it shows
The graph of y = ax² + bx + c is a parabola whose shape and position follow from its coefficients. The sign of a sets the opening: upward with a minimum value at the vertex when a > 0, downward with a maximum when a < 0, and a larger |a| gives a narrower curve. The vertex is I(−b/2a; −Δ/4a) with Δ = b² − 4ac, the axis of symmetry is x = −b/2a, the y-intercept is (0; c), and the sign of Δ decides whether the graph crosses the x-axis twice, touches it or misses it.
How to use
Move Coefficient a, Coefficient b and Coefficient c one at a time and ask students to describe how the vertex and the axis of symmetry respond before reading the numbers. Dragging vertex I on the graph changes b and c directly. Keep Show variation table on to link the arrows to the sign of a, and Show intercepts with both axes on to connect Δ with the number of x-intercepts.
Parameters you can change
- Coefficient a -3–3
- Coefficient b -8–8
- Coefficient c -8–8
- Show variation table
- Show intercepts with both axes
Questions to explore
- If only c changes, how does the vertex move, and why does the axis of symmetry stay where it is?
- What is the maximum value of y = −x² + 4x, and at which value of x is it reached?
- For which value of c does y = x² − 4x + c have exactly one x-intercept?
Teaching ideas
All guides- Interactive Math Activities with SimulationsInteractive math activities for grades 4–12: ten simulation tasks on fractions, ratios, integers, equations, graphs, trigonometry, geometry and statistics.
- Quadratic Functions Lesson Plan with Interactive GraphsA quadratic functions lesson plan with interactive graphs: what a, b and c do, vertex and roots with expected values, vertex form, and a five-question set.
