Turning points and roots

You can read these from the graph of a quadratic:

• Roots: where it crosses the x-axis (A and B below). They are the solutions of the equation = 0.
• y-intercept: where it crosses the y-axis (C).
• Turning point: the lowest point of a U, or the highest of an ∩ (T).
• Line of symmetry: the vertical line through the turning point.

Example: use the graph of y = x² − 2x − 3 to find its roots, y-intercept, turning point and line of symmetry.

1. Roots: x = −1 and x = 3
2. y-intercept: (0, −3)
3. Turning point: (1, −4)
4. Line of symmetry: x = 1

The line of symmetry is exactly halfway between the roots, so you can find the turning point without a graph.

1. Find the roots by factorising.
2. The line of symmetry is x = (first root + second root) ÷ 2.
3. Substitute that x to find the y-coordinate of the turning point.

Example: find the turning point of y = x² + 2x − 8.

1. x² + 2x − 8 = (x + 4)(x − 2), so the roots are x = −4 and x = 2
2. Line of symmetry: x = (−4 + 2) ÷ 2 = −1
3. y = (−1)² + 2 × (−1) − 8 = 1 − 2 − 8 = −9
4. Turning point (−1, −9)

The y-intercept is the number on its own: (0, −8).

y = (x + p)² + q has its turning point at (−p, q). The squared bracket is never negative, so it is smallest (0) when x = −p.

Example: find the turning point of y = x² − 6x + 11.

1. (x − 3)² − 9 + 11 = (x − 3)² + 2
2. Minimum point (3, 2), above the x-axis, so there are no roots

Harder example: find the turning point of y = −x² + 4x + 1.

1. Take out −1: −(x² − 4x) + 1
2. x² − 4x = (x − 2)² − 4, so −[(x − 2)² − 4] + 1 = −(x − 2)² + 5
3. Maximum point (2, 5)