The 3 Standard Forms of Quadratic Functions
A quadratic function is a second-degree polynomial equation. Depending on the problem, it can be written in three equivalent forms:
- Standard Form:
y = ax² + bx + c(Best for quadratic formula and y-intercept) - Vertex Form:
y = a(x - h)² + k(Best for identifying maximum/minimum vertex (h, k)) - Factored (Intercept) Form:
y = a(x - p)(x - q)(Best for finding x-intercepts p and q)
The Discriminant: Predicting the Nature of Roots
The term inside the square root of the Quadratic Formula is called the Discriminant (Δ):
Δ = b² - 4ac
| Discriminant Value | Number & Type of Roots | Parabola Graph Interaction |
|---|---|---|
| Δ > 0 | 2 distinct real roots | Crosses the X-axis at two distinct points |
| Δ = 0 | 1 repeated real root | Touches the X-axis at its vertex (tangent) |
| Δ < 0 | 2 complex conjugate roots | Never touches or crosses the X-axis |
How to Convert Standard Form to Vertex Form
To convert y = ax² + bx + c to vertex form y = a(x - h)² + k:
- Calculate h = -b / (2a)
- Calculate k = f(h) = ah² + bh + c
- Substitute a, h, and k into
y = a(x - h)² + k