Quadratics - the Australian Mathematical Sciences Institute

{8} • Quadratics
Reflections
Parabolas can also be reflected in the x-axis. Thus the parabola y = −x 2 is a reflection of
the basic parabola in the x-axis.
y
y
y = x2
(a,a2)
y = –x2
0
a
x
(a,–a2)
0 a
x
Example
Find the vertex of y = −x 2 + 6x − 8 and sketch its graph.
Solution
Completing the square, we have
y = − ( x 2 − 6x + 8 )
= − ( (x − 3) 2 − 1 ) = 1 − (x − 3) 2 .
Hence the vertex is (3,1) and the parabola is ‘upside down’. The equation of the axis of
symmetry is x = 3. The graph is shown below.
y
1
y = –x2 + 6x – 8
= 1 – (x–3)2
0
2
3 4
x