What kind of a visual trajectory would an asteroid have if it were to hit Earth?
There is no simple answer to this question other than: you have to do the math.

Most often you would see the asteroid brightening more rapidly and moving across the sky faster as the days pass, but not necessarily.  It could arrive directly along our orbital path and just brighten but show no motion in the sky.  It could start out lit up fully (full phase) and then arrive with only partial lighting from the sun, so it does not necessarily brighten.   

You generally would not know the phase, unless you calculate it, because you will not be able to resolve it until it gets really close.