Any addition or subtraction represents a shift. So in this case you have +4 and a -8. Any multiplication represents some sort of stretch or compression. If the multiplication is by a negative value it will also be a reflection.
The way I kind of explain it is to say is your value "trapped" with x or not? As in can you freely move that number around. So examples of "trapped" are
\((x-1)^{2}\)
\(\sqrt{x-6}\)
\(|x+2|\)
Where your numbers are being bounded by some sort of grouping symbol.
So, if your transformations are not bound by a grouping symbol, they will affect y-values. So for us, the -3 multiplication and the -8 subtraction are applied to y-values. The +4 is bounded and trapped with x, so it will be the only thing affecting x-values.
Now, transformations that affect y-values do as they look like they might do. If it's a +3 shift, things will go up 3. If it's a multiplication by 2, all the y-values will be multiplied by 2. Now the transformations that affect x-values, the "trapped" ones, kind of do the opposite. If it's a +5, you go left 5, more into the negative values for x. And if you multiply by 2, all the x-values shrink by a factor of 2 (or are multiplied by 1/2 you can say).
So after all that, let's put everything together. Now, we want to consider multiplicative transformations first. The only one of those is the -3. So all the y-values of x^6 were multiplied by -3. In terms of transformations, this represents a reflection about the x-axis and a stretch by a factor of 3. After that, the graph was shifted 4 to the left and down 8.