Power of Maths: Paper 1 – Section 7 4. Finding the axis of symmetry
The axis of symmetry of a quadratic function y = ax2 + bx + c is the line through the vertex V parallel to the y-axis. y
V
P( p, 0)
x =
— – 2a
b
Q(q, 0)
x
The equation of the axis of symmetry is x = − b ___
This means the roots of ax 2 + bx + c = 0 are symmetrical about x = − b ___
2a . 2a . P( p, 0)
( − b ___
V – b
— 2a, 0
2a , 0 ) is the midpoint of P( p, 0) and Q(q, 0).
EXAMPLE 11 For the quadratic function y = x 2 – 2x – 8, fi nd the point V which gives the function its minimum value. Find the roots of x 2 – 2x – 8 = 0 and show they are symmetrical about the x co-ordinate of V.
Solution y = x 2 – 2x – 8
a = 1, b = –2, c = –8 Vertex V: x = − b
___ 2a = 2
__ 2 = 1
x = 1: y = (1)2 – 2(1) – 8 = –9 Therefore, the x co-ordinate of the vertex V (1, –9) is 1. Roots: x2 – 2x – 8 = 0 (x + 2)(x – 4) = 0 x = –2, 4
