Quadratic Function and the Parabola
The quadratic function f(x) = ax² + bx + c has a parabola as its graph. The coefficient a determines the concavity and the opening; the vertex is the maximum or minimum point.
Discriminant: Δ = b² − 4ac Vertex: xᵥ = −b/(2a) yᵥ = −Δ/(4a) Roots (quadratic formula): x = (−b ± √Δ) ÷ (2a) a > 0 → opens upward (minimum at yᵥ) a < 0 → opens downward (maximum at yᵥ)
Practical applications
- Physics: projectile trajectory — h(t) = −gt²/2 + v₀t + h₀
- Economics: profit maximization — P(x) = −ax² + bx − c
- Geometry: sizing plots of land with maximum area and fixed perimeter
- Engineering: parabolic arches in bridges and viaducts
Frequently asked questions
Why is a ≠ 0 in a quadratic function?
If a = 0, the expression ax² + bx + c reduces to bx + c (a linear function). The quadratic term x² is what defines degree 2 and the parabola shape.
What does Δ < 0 mean on the graph?
The parabola does not cross the x-axis. If a > 0 it lies entirely above the x-axis (f(x) > 0 for every x). If a < 0 it lies entirely below.
How do you use the quadratic function to maximize area?
Write the area as a function of one variable, identify a, b, c and compute the vertex. The coordinate xᵥ gives the optimal dimension.
How are quadratic functions and quadratic equations related?
The zeros of f(x) = ax² + bx + c are the roots of the equation ax² + bx + c = 0. They are two views of the same mathematical object.