What is the range of a quadratic function?
1 Answer
The range of
#{ ([c-b^2/(4a), oo) " if " a > 0), ((-oo, c-b^2/(4a)] " if " a < 0) :}#
Explanation:
Given a quadratic function:
#f(x) = ax^2+bx+c" "# with#a != 0#
We can complete the square to find:
#f(x) = a(x+b/(2a))^2+(c-b^2/(4a))#
For real values of
Then:
#f(-b/(2a)) = c - b^2/(4a)#
If
If
Another way of looking at this is to let
Given:
#y = ax^2+bx+c#
Subtract
#ax^2+bx+(c-y) = 0#
The discriminant
#Delta = b^2-4a(c-y) = (b^2-4ac)+4ay#
In order to have real solutions, we require
#(b^2-4ac)+4ay >= 0#
Add
#4ay >= 4ac-b^2#
If
#y >= c-b^2/(4a)#
If
#y <= c-b^2/(4a)#