Quadratic graphs and their roots

Years 10 to 12

A quadratic makes a U-shaped curve, and its roots are where it crosses the x-axis.

y = x² − 4x + 3. It crosses the x-axis at 1 and 3, and has its lowest point at (2, −1).

For learners

A quadratic equation has an $x^{2}$ in it, like $y = x^{2} - 4 x + 3$ . Its graph is a smooth U-shape called a parabola.

The roots are the x-values where the curve crosses the x-axis. For this curve they are 1 and 3.

The curve is the same on both sides of its lowest point. The line down the middle is the line of symmetry.

A worked example

Solve $x^{2} - 5 x + 6 = 0$ .

Factorise: $(x - 2) (x - 3) = 0$ .
So $x - 2 = 0$ or $x - 3 = 0$ .
$x = 2$ or $x = 3$ .

For grown-ups

Factorising $x^{2} - 4 x + 3 = (x - 1) (x - 3)$ gives the roots directly. The turning point lies halfway between the roots, and completing the square gives it as $(x - 2)^{2} - 1$ , so the lowest point is $(2, - 1)$ .

Common mistakes

Writing the roots with the wrong sign: the factor $(x - 1)$ gives root $+ 1$ .
Forgetting that a negative $x^{2}$ term makes the graph turn upside down.
Not checking the answer by substituting back in.

Why it matters

Quadratics describe the path of a thrown ball, areas, and many problems in physics and engineering.

Topics in quarter that use this

Solving quadratics
Factorising quadratics
Completing the square
The quadratic formula
Quadratic graphs