Originally Posted by

**Reckoner** If a function is increasing over an interval, its derivative will be positive, and if it is decreasing it will be negative. I assume you know how to do differentiation, since you posted in calculus.

Of course, you can also use the equation of the parabola to determine intervals on which it is increasing or decreasing. In standard form, a parabola can be represented by the equation

$\displaystyle y - k = \frac1{4p}(x - h)^2$

where $\displaystyle (h,\;k)$ is the vertex and $\displaystyle p$ is the directed distance between the vertex and the focus. If $\displaystyle p$ is positive, the parabola opens up, and if negative, the parabola opens down.

Since your parabola, $\displaystyle f(x) = x^2 + 2 = (x - 0)^2 + 2$ has a vertex at $\displaystyle (0,\;2)$ and opens up, you can conclude that it is decreasing for $\displaystyle x\in(-\infty,\;0)$ and increasing for $\displaystyle x\in(0,\;\infty)$.