Quadratic Functions
Master parabolas, vertex form, and the quadratic formula. Quadratic functions appear in physics, engineering, and data modeling whenever quantities change at a non-constant rate.
Learning Path
Quadratic Functions Poster
Visual guide to quadratic functions in standard, vertex, and factored form. Covers the quadratic formula, discriminant, vertex, and axis of symmetry.
Open →Quadratic Equation Solver
Solve ax^2 + bx + c = 0 with an interactive graph. See roots, vertex, discriminant, and step-by-step solutions using factoring, completing the square, or the quadratic formula.
Open →Quadratic Functions and Parabolas Lab
Explore standard, vertex, and intercept forms interactively. Find the vertex, axis of symmetry, roots, and discriminant. Complete the square and compare all three forms with an interactive graph.
Open →Key Formulas
The essential quadratic formulas at a glance.
- Standard: y = ax^2 + bx + c
- Vertex: y = a(x - h)^2 + k
- Quadratic formula: x = (-b +/- sqrt(b^2 - 4ac)) / 2a
- Discriminant: D = b^2 - 4ac
- Vertex x-coordinate: h = -b / (2a)
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Common Questions
What does the discriminant tell you about the roots?
The discriminant D = b^2 - 4ac tells you the nature of the roots. If D > 0 there are two distinct real roots; if D = 0 there is exactly one real root (a repeated root); if D < 0 there are two complex (non-real) roots.
When should you use vertex form instead of standard form?
Vertex form y = a(x - h)^2 + k is most useful when you need to identify the vertex (h, k) and axis of symmetry directly, or when graphing transformations. Standard form is convenient for applying the quadratic formula.