The range of a quadratic function

A quadratic graph is a ∪ or a ∩. It's a ∪ when the $x^2$ term is positive, and a ∩ when it's negative.

Its most extreme output (lowest for ∪, highest for ∩) is at the turning point, called the vertex.

Completing the square hands you the vertex for free. Take $f(x) = (x - 3)^2 + 2$:

• $(x - 3)^2$ is a square, so it's never negative. Its smallest value is 0, when $x = 3$.
• So the smallest value of $f(x)$ is $0 + 2 = 2$. The vertex is $(3, 2)$.

In general, with $a > 0$:

Why $-p$? The bracket $(x + p)$ is zero when $x = -p$. So $(x - 3)^2$ has $p = -3$ and the vertex is at $x = 3$.