# 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)#