F(x) = (x + 6)2 + 3; horizontal shrink by a factor of 1/2 and a translation 1 unit down, followed by a reflection
in the x-axis

A function s is the HORIZONTAL SHRINK of a function h by a factor k

if s(x) = h(kx)

It is called a shrink, because for k > 1 that has the effect of compressing

the function h towards the y-axis.

I find the problem statement a little confusing because

using k = 1/6 would result in stretching, not shrinking.

Therefore my guess is that the author means k = 6.

A function r is a REFLECTION of a function h in the x-axis

if r(x) = -h(x).

The reason is that r looks like a mirror image of h,

where the mirror is the x-axis  

A function t is a TRANSLATION 2 units down of a function h

if t(x) = h(x) - 2.

The graph of t looks like h, except shifted down 2 units.

Applying the definitions to your function f:

g(x) = -f(6x) - 2 = -36x^2 - 2