Her answer is correct, but she used logarithms.  Maybe you aren't supposed to
use logarithms. Maybe you're not studying logarithms yet.  Anyway, here's
how without using logarithms. 

Rewrite y=4e^{-0.5t} in the form y=a(1+r)^t 

   

If those are to be identical, then regardless of what number we choose
to substitute for t, both sides of that equation must be equal.

The easiest number to substitute is 0.

   

   

Since anything raised to the zero power is 1 [With one exception, the undefined
00, which is of no concern to us here.]

So the above equation becomes:

   
     

So "a" is 4.  So we substitute 4 for a:

   

We divide both sides by 4

   

Now we substitute t=1

   

   

   

   

You can rewrite r as



which she did.

So to write 

y=4e^{-0.5t} in the form y=a(1+r)^t is to write it



And that simplifies to her answer, which is correct but not quite in
the form you asked for.  

Edwin