# How do you find domain and range of a quadratic function?

##### 2 Answers

The domain is all real numbers and the range is all the reals at or above the vertex y coordinate (if the coefficient on the squared term is positive) or all the reals at or below the vertex (if said coefficient is negative).

See explanation...

#### Explanation:

Suppose:

#f(x) = ax^2+bx+c \ # where#a != 0#

First note that

We can complete the square and find:

#f(x) = a(x+b/(2a))^2+c-b^2/(4a)#

This is vertex form for a parabola with vertex at

Note that for real values of

#(x+b/(2a))^2 >= 0#

with equality when

Hence if

In fact for any

#f((-b+-sqrt(b^2-4a(c-y)))/(2a)) = y#

So the range of

Similarly if

#f((-b+-sqrt(b^2-4a(c-y)))/(2a)) = y#

So the range of