Expert: Paul Klarreich - 2/24/2006

Question
Hello.
How do you find the range of a function when it is in completed square form- (x+a)^2 +b

I took the equation:
x^2 -6x +5
and put it into completed square form:
(x-3)^2 -4
But how do you know find the range of the fuction?

Thank you.

Answer
Hi, Simon,

You wrote:
Subject:  Range
Question:  Hello.
How do you find the range of a function when it is in completed square form- (x+a)^2 +b

I took the equation:  x^2 -6x +5
and put it into completed square form:
(x-3)^2 -4
But how do you know find the range of the function?

Thank you.
--------------------------------------
You did well.  Now that you have

y = (x-3)^2 - 4

you observe that:

A. (x-3) is the square of a number.
B. The square of a number cannot be negative, so its smallest value is zero.  So (x-3)^2 >= 0

C. But you have an extra term, -4.  Add that to the inequality:

   (x - 3)^2      >= 0
             - 4       - 4
-----------------------------
   (x - 3)^2 - 4  >= -4

D. But that left side is y.  So your range is  y >= -4.