Intervals of Increase and Decrease: Analyzing y = (x - 12.5)² - 4

To find intervals where the function is increasing or decreasing, follow these steps:

• The given function is $y=\left(x-12\frac{1}{2}\right)^2-4$.
• This is a parabola in the form $y=(x-h)^2 + k$, with vertex $(h, k)$.
• Identify the vertex: $h = 12.5$, $k = -4$.
• The vertex indicates the minima for this parabola since it opens upwards.

For quadratics like this:

• It decreases on the interval to the left of the vertex: $x < 12.5$.
• It increases on the interval to the right of the vertex: $x > 12.5$.

Thus, the function $y=\left(x-12\frac{1}{2}\right)^2-4$ is decreasing for $x < 12.5$ and increasing for $x > 12.5$.

The correct choice, matching this analysis, is choice 4: $\searrow:x < -12\frac{1}{2}\\\nearrow:x > -12\frac{1}{2}$

Therefore, the intervals of increase and decrease are:

$\searrow:x < -12\frac{1}{2}, \nearrow:x > -12\frac{1}{2}$.