Figure 11-2: Graph of function y = x2 - 2x - 3
The maximum or minimum of a quadratic occurs at the vertex of the graph. By completing the square, any quadratic function can be written as:
In this form, the location of the vertex is given by {h, k}. For the function:f(x) = a(x - h)2 + k
the vertex is: {-3, -2}f(x) = (x + 3)2 - 2
Maximum and minimum values
The extreme value of a quadratic is the value of the function at the vertex. In the vertex form, the extreme value occurs when x = h and is equal to k. When a > 0 is true, the extreme value is a minimum; when a < 0 is true, it is a maximum.
Completing the square can be done symbolically to give equations for the vertex of a parabola:
f(x) = ax2 + bx + c
= a[x2 + (b/a)x] + c
= a{x2 + (b/a)x + [b/(2a)]2} - a[b/(2a)]2 + c
= a[x + b/(2a)]2 + [(4ac - b2)/(4a)]
Then we get:
-b
h = ---- axis of symmetry
2a
4ac - b2
k = -------- extreme value
4a
Example, find the vertex of f(x) = x2 - 2x - 3
Then the vertex is: {1, -4}h = 2/2 = 1 k = [4×(-3) - 4]/4 = -16/4 = -4
Figure 11-2: Graph of function y = x2 - 2x - 3