# How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for  f(x) = x² + 2x - 3?

Use the derivative. The function is (strictly) increasing over intervals where $f ' \left(x\right) > 0$ and (strictly) decreasing over intervals where $f ' \left(x\right) < 0$. Local extreme points occur at critical points where $f ' \left(x\right) = 0$ or where $f ' \left(x\right)$ is undefined.
If $f \left(x\right) = {x}^{2} + 2 x - 3$, then $f ' \left(x\right) = 2 x + 2$ so that $f ' \left(x\right) < 0$ when $x < - 1$ and $f ' \left(x\right) > 0$ when $x > - 1$. This means $f$ is decreasing over the interval $x \le - 1$ and $f$ is increasing over the interval $x \ge - 1$. $f$ has a local minimum value at the critical point $x = - 1$ equal to $f \left(- 1\right) = 1 - 2 - 3 = - 4$.