The asymotopes wnt be affected, in that example by compression and expansion as the asymotopes lie on the x and y axis, the general steepness of the graph will increase tho.
If you added a shift to it the expansion or compression would affect the asymotopes just as it would any other point
if you had y=1/x +2 where the asymotopes are at y=2 and x=0
a vertical expansion of scale factor 2... y=2(1/x +2) can be written as y=2/x +4. Where the asymotopes are y=4 and x=0
note: y=1/x +1 is not equal to y=1/(x+1) it wasn't very clear wen i wrote it.
But consider , say. This can be got by applying appropriate translations to f(x). g(x) has a vertical asymptote x = 1 and a horizontal asymptote y = 2. (So applying translations to will change its asymptotes, if that's what you were asking).
But if you want to know whether dilations will have an affect on asymptotes after translations have been added to , then consider:
Dilation by factor a from horizontal axis: . Clearly .
h(x) still has a vertical asymptote x = 1 but the horizontal asymptote is y = 2a.
Dilation by factor 1/a from vertical axis: . Clearly .
k(x) now has a vertical asymptote x = 1/a and the horizontal asymptote is still y = 2.