Understanding Vertex Form and Its Expansion
The vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex and a determines the parabola's width and direction. To derive the standard form y = Ax^2 + Bx + C, expand the vertex form. Start by expanding (x - h)^2 to x^2 - 2hx + h^2, then multiply by a: y = a(x^2 - 2hx + h^2) + k = Ax^2 + Bx + C, where A = a, B = -2ah, and C = ah^2 + k. This directly answers how to derive the equation by algebraic expansion.
Key Principles of the Derivation
The derivation relies on the binomial theorem for expanding (x - h)^2 and distributing the coefficient a. The vertex (h, k) shifts the parabola from the origin, and expanding reveals the quadratic, linear, and constant terms. This process preserves the parabola's properties, such as its axis of symmetry at x = h, and allows conversion between forms for graphing or solving problems.
Practical Example: Deriving from Vertex Form
Consider the vertex form y = 2(x - 3)^2 + 1, with vertex (3, 1) and a = 2. Expand: (x - 3)^2 = x^2 - 6x + 9. Multiply by 2: 2x^2 - 12x + 18. Add k: y = 2x^2 - 12x + 19. Thus, A = 2, B = -12, C = 19. This example illustrates applying the steps to find the standard form, useful for completing the square in reverse or analyzing intercepts.
Importance and Real-World Applications
Deriving the standard form from vertex form is essential for quadratic modeling in physics (e.g., projectile motion) and engineering (e.g., parabolic arches). It enables easy calculation of roots using the quadratic formula or vertex coordinates for optimization problems, bridging theoretical algebra with practical applications like trajectory predictions.