Sketch y = ax² + bx + c
A standard method from Functions and graphs: the steps and worked examples.
Steps
- Direction: a > 0 opens up (has a minimum); a < 0 opens down (has a maximum).
- y-intercept: c.
- x-intercepts: solve ax² + bx + c = 0 (if Δ ≥ 0).
- Axis of symmetry: x = −b/(2a), or halfway between the x-intercepts.
- Vertex: put that x into the function.
Worked examples
Example 1: Sketch y = f(x) for f(x) = x² − 6x + 5: find f(−1), the intercepts and the vertex.
f(−1) = 1 + 6 + 5 = 12
y-intercept: 5
x-intercepts: (x − 1)(x − 5) = 0, so x = 1, 5
axis: x = 3 (halfway between 1 and 5)
vertex: f(3) = 9 − 18 + 5 = −4, so (3, −4)For the whole topic, read Functions and graphs: key points, common mistakes and how the DSE tests it.