Key Idea: A quadratic graph is a parabola, and you can write it three ways — each hands you a different feature for free. It's mostly Paper 1 (by hand): sketch it, find the roots, or find the turning point.
📐 The three forms — what each reveals
| Form | Looks like | Read off instantly |
|---|---|---|
| Standard | y = ax² + bx + c | Direction (sign of a) and the y-intercept c, i.e. (0, c). |
| Vertex | y = a(x − h)² + k | The turning point (h, k) — and k is the max/min value. |
| Factored | y = a(x − p)(x − q) | The x-intercepts x = p and x = q (set each bracket to 0). |
a > 0 → opens up → minimum. a < 0 → opens down → maximum. Watch the bracket signs: (x + 1) gives the root x = −1, and (x − 3)² gives h = +3.
🎯 Vertex & axis of symmetry
- the coefficients in y = ax² + bx + c
- the axis of symmetry — also midway between the two x-intercepts; the vertex sits on it
Tip: To reach a(x − h)² + k: halve the x-coefficient, square it for the bracket, then add/subtract to fix the constant. The vertex is (h, k), and k is the min (a > 0) or max (a < 0) value, since the squared part is never negative.
✏️ IB-style worked examples
IB-style question — direction & y-intercept from standard form
For y = −3x² + 4x − 7, state the direction it opens and its y-intercept.
Step by step:
The sign of a sets the direction.
The constant c is the y-intercept.
Opens downward (so the vertex is a maximum); y-intercept (0, −7).
IB-style question — find the x-intercepts (factored form)
Find the x-intercepts of y = (x − 5)(x + 2).
Step by step:
Set each factor equal to zero.
Solve each — mind the sign on the second bracket.
x-intercepts at (5, 0) and (−2, 0).
IB-style question — write in the form (x − h)² + k
Write x² − 8x + 19 in the form (x − h)² + k, and state the vertex.
Step by step:
Halve the x-coefficient (−8 → −4) and square it (16).
Fix the constant: 19 = 16 + 3.
(x − 4)² + 3, so the vertex is (4, 3).
IB-style question — build a quadratic from its vertex
A parabola has vertex (3, −4) and passes through (1, 4). Find a in y = a(x − 3)² − 4.
Step by step:
Substitute the known point (1, 4).
Simplify and solve for a.
a = 2, so y = 2(x − 3)² − 4.
Important: In a(x − h)² + k, the vertex x-coordinate is +h: (x − 3)² means h = 3, not −3. And re-read what's asked: the min/max value is k (a number); the min/max point is (h, k).
Tap each card to reveal the answer.
Which form gives the y-intercept at a glance? Standard form y = ax² + bx + c — the y-intercept is (0, c).
Roots of y = (x − 6)(x + 4)? x = 6 and x = −4 — set each bracket to zero.
Axis of symmetry of y = 2x² − 12x + 1? x = 3 — use x = −b/(2a) = −(−12)/(2·2).
Vertex of y = (x + 5)² − 2? (−5, −2) — (x + 5)² gives h = −5, and k = −2.
Does y = −x² + 6x − 1 have a max or a min? A maximum — a = −1 < 0, so the parabola opens down.
Exam tips
- Pick the form that answers the question: factored → roots, vertex → turning point, standard → y-intercept & direction.
- Sign of a: a > 0 opens up (minimum), a < 0 opens down (maximum).
- Axis of symmetry x = −b/(2a) is also exactly midway between the two x-intercepts.
- Complete the square: halve b, square it, then add/subtract to keep the constant correct.
- (x − h)² gives h = +h; the min/max value is k, the min/max point is (h, k).