Constructing the canonical form

Why use a different form?

The expanded form f(x) = ax² + bx + c is useful for calculating an image, but it hides the vertex of the parabola. The canonical form is written as:

f(x) = a(x - alpha)² + beta

where alpha = -b/(2a) and beta = f(alpha).

Method of calculation

We start with f(x) = ax² + bx + c and express x² + b/a x as a perfect square:

1. We factor out a: f(x) = a(x² + b/a x) + c
2. We complete the square: x² + b/a x = (x + b/(2a))² - b²/(4a²)
3. We simplify to obtain f(x) = a(x - alpha)^2 + beta

Example

Let f(x) = 2x^2 - 8x + 5.

• a = 2, b = -8, c = 5
• alpha = -b/(2a) = 8/4 = 2
• beta = f(2) = 2*4 - 16 + 5 = -3

Therefore, f(x) = 2(x - 2)^2 - 3.

Common pitfall

Don’t forget to factor by a before completing the square: if a ≠ 1, you must first factor out a as a common factor from the x² and x terms; otherwise, the calculation of alpha will be incorrect.