IP EMaths Notes (Upper Sec, Year 3-4): 06) Quadratic Functions and Graphs

Quadratic graphs model projectile paths, profit curves, and optimisation problems. Completing the square reveals turning points instantly and helps evaluate maxima or minima.

Toolkit

• Standard form: \(y = ax^{2} + bx + c\).
• Completed square: \(y = a\left(x - h\right)^{2} + k\) where \((h, k)\) is the vertex.
• Axis of symmetry: \(x = h\). Discriminant \(\Delta = b^{2} - 4ac\) indicates intercept behaviour.

Worked example - Sketch from completed square

Sketch \(y = x^{2} - 6x + 5\), stating the axis of symmetry and intercepts.

1. Complete the square: \(x^{2} - 6x = (x - 3)^{2} - 9\). So \(y = (x - 3)^{2} - 4\).
2. Vertex: \((3, -4)\). Axis of symmetry: \(x = 3\).
3. \(y\text{-intercept}\): set \(x = 0 \implies y = 5\).
4. \(x\text{-intercepts}\): solve \(x^{2} - 6x + 5 = 0\). Factor: \((x - 1)(x - 5) = 0 \implies x = 1\) or \(x = 5\).
5. Sketch using axis, vertex, and intercepts.

Try this

Rewrite \(y = 2x^{2} + 8x + 3\) in completed-square form and state whether the curve has a minimum or maximum value.