You convert between quadratic forms by expanding, factorizing, or completing the square. Each form represents the same parabola but makes different features immediately visible.
This is IB Mathematics: Analysis and Approaches sub-topic SL 2.6: Quadratic function, applicable to both AA SL and HL.
The Three Forms
| Form | Equation | Feature shown directly |
|---|---|---|
| General form | The -intercept is | |
| Factorized form | The roots are and | |
| Vertex form | The vertex is ((h,k)) |
In every form, is the same and determines whether the parabola opens upward or downward.
The value of also controls vertical stretch: larger (|a|) produces a narrower parabola, while produces a wider one. The sign of determines orientation, so opens upward and opens downward. Because , every quadratic can be written in general and vertex form, although factorized form over the real numbers exists only when real roots exist.
Conversion Methods
| Conversion | Method |
|---|---|
| Factorized to general | Expand the brackets and collect like terms. |
| Vertex to general | Expand the square and simplify. |
| General to factorized | Factorize, or find the roots and write (a(x-p)(x-q)). |
| General to vertex | Complete the square, or calculate and . |
For example, begin with
Factorizing gives
Completing the square gives
Therefore, the same quadratic has roots and , vertex , and -intercept .
The axis of symmetry is . In factorized form, it lies halfway between the roots, which explains why when both roots are real.
A common misconception is that changing form changes the function. It does not: the algebraic expression changes, but the graph remains identical.
Exam Technique
For a show or find question, include the algebraic stages rather than writing only the converted form. On Paper 1, expect exact factorization or completing the square; always check your conversion by expanding back to general form.