Quadratic Visualiser

Mathematics module exploring how the coefficients in y = ax^2 + bx + c control the shape, roots, and turning point of a parabola.

Learning objectives

• Identify the role of coefficients a, b, and c in a quadratic function.
• Explain how the sign and magnitude of a affect opening direction and steepness.
• Use the discriminant to classify real roots.
• Calculate and interpret the vertex as the turning point of the parabola.
• Connect algebraic results to graph behaviour.

Mathematical background

A quadratic function has the standard form y = ax^2 + bx + c, where a must be non-zero. The graph is a parabola.

• The coefficient a controls whether the parabola opens upward or downward.
• The coefficient b influences the horizontal position of the vertex.
• The coefficient c gives the y-intercept.
• The discriminant D = b^2 - 4ac determines whether real roots exist.
• The vertex occurs at x = -b / (2a), with y found by substituting that x-value into the function.

Core equations

• y = ax^2 + bx + c
• D = b^2 - 4ac
• x = (-b +/- sqrt(D)) / (2a), when D is non-negative
• x_vertex = -b / (2a)
• y_vertex = a(x_vertex)^2 + b(x_vertex) + c

Interactive simulator

Launch locally on port 8504.

streamlit run simulations/quadratic_visualiser/streamlit_quadratic_app.py --server.port 8504

Worked example

For y = x^2 - 5x + 6, the discriminant is 1, so there are two real roots. The roots are x = 2 and x = 3, and the vertex lies midway between them at x = 2.5.

Engineering meaning

Quadratic models appear in optimisation, projectile paths, curved structural profiles, cost modelling, and many systems where a quantity changes with the square of another variable.

Challenge questions

1. What changes when a is negative instead of positive?
2. Why does D = 0 mean the curve touches the x-axis only once?
3. How does changing c move the graph?

HeliosSTEMLab · Quadratic Visualiser Recovery module