The same parabola, written three ways: Every quadratic can be written in three forms, and each hands you a different feature for free.
Standard
- c = y-intercept
- sign of a = direction
Factored
- p, q = x-intercepts
- the roots / zeros
Vertex
- (h, k) = vertex
- max / min value k
Choose the form that answers the question: Want the roots?
Use factored.
Want the turning point?
Use vertex.
Want the y-intercept?
Use standard.
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Read a and c at a glance: In ax² + bx + c: the sign of a sets the direction (a > 0 opens up, a < 0 opens down), and c is the y-intercept (where x = 0).
IB-style question — read it off
For y = −2x² + 3x − 5, state the direction it opens and its y-intercept.
Step by step
- a = −2 < 0, so it opens downward.
- c = −5 is the y-intercept.
Final answer
Opens downward; y-intercept (0, −5).
Opening down means a maximum: a < 0 → opens down → the vertex is a maximum. a > 0 → opens up → the vertex is a minimum.
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Each bracket gives a root: In a(x − p)(x − q), the x-intercepts are x = p and x = q — set each bracket to zero.
Watch the signs: (x + 1) gives the root x = −1.
IB-style question — roots from factors
Find the x-intercepts of y = (x − 4)(x + 1).
Step by step
- Set each factor to zero.
- Solve.
Final answer
x-intercepts at (4, 0) and (−1, 0).
The vertex sits midway: By symmetry, the axis of symmetry is halfway between the roots — here at x = (4 + (−1))/2 = 1.5 (more on this in 2.6.2).
Direction + intercepts + vertex: To sketch a quadratic: get the direction (sign of a), the x-intercepts (factor), the y-intercept (c), and the vertex — then draw the smooth U or ∩ through them.
IB-style question — sketch a parabola
Sketch y = x² − 2x − 3.
Step by step
- Direction: a = 1 > 0, opens up.
- Factor for the roots.
- y-intercept and vertex.
Final answer
Upward parabola through (−1, 0), (3, 0), (0, −3), with minimum (1, −4).
The sketch of y = x² − 2x − 3: roots at x = −1 and x = 3, y-intercept −3, and minimum vertex (1, −4).
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