The graph of a quadratic function touches, but does not cross, the x-axis at x = 4. What function represents this situation?

Since it has only one zero when x=4, we can say that the quadratic results from a single factor squared, namely ...

(x-4) ^2 which if you wish to expand is:

f (x) = x^2-8x+16

However, note that the questioner may not be looking for or expecting it, the negative of f (x) above has the single zero at x=4 as well, but opens downwards instead of upwards ...

f (x) = - x^2+8x-16