Find the turning point.

A quadratic contains an $x^{2}$ term and its graph is a parabola. Factorisation exposes its roots: if (x − r)(x − s) = 0, either x = r or x = s. The zero-product rule applies only when the product equals zero.

Vertex form y = a(x − h)² + k gives a turning point at (h, k) and symmetry line x = h. Positive a opens upward; negative a opens downward. Expanding the brackets connects this form with ax² + bx + c.

INTERACTIVE MODEL

Try it. Watch it change.

y = 1(x − 0)² + 0

Vertical scale / gradient a1 Horizontal shift h0 Vertical shift k0Input x1

f(1) = 1 · Vertex (0, 0)

The graph is calculated from the displayed rule. Drag the highlighted point or use the sliders.

Explore: Adjust a, h and k. Make a parabola with a maximum at (1, 3).

Turning-point form

y=a(x-h)^2+k

WORKED EXAMPLE

Find the roots and minimum of y = $x^{2}$ − 6x + 8.

Factorise: y = (x − 2)(x − 4), so the roots are 2 and 4.
Complete the square: y = (x − 3)² − 1.
The minimum is −1 at x = 3.

Assumed knowledge

Expanding brackets and multiplying signed numbers.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Factorise and solve monic quadratics.
Connect roots, symmetry and vertex form.

Year 10 preparation ↗

READY TO TRY IT?

Find the turning point.

Find the roots and minimum of y = x2x^{2}x2 − 6x + 8.

Put the idea to work.

Find the roots and minimum of y = $x^{2}$ − 6x + 8.