Given vertical translation a (+ to shift up, - to shift down), vertical scaling b (larger to grow, smaller to shrink), horizontal translation c (+ to shift right, - to shift left), and horizontal scaling d (larger to grow, smaller to shrink), the function transformation would be:
You can pick and choose values. If you don't have a or c, replace with 0. If you don't have b or d, replace with 1.
Assume our function f was the sine function. sin(x):
Say we wanted to shift the whole function to the right by +1, i.e. up the x-axis. c=1 and our transformation would be sin(x - [+1]):
Note how the graph used to cross the x-axis at 0 (red), but now crosses it at 1 (green).
Say we now wanted to stretch it longer, i.e. horizontally scale it. Maybe stretch it twice as long. If we choose a horizontal scaling factor of d=2, our new function would be sin((x-1)/2):
Now perhaps we want to shift it upwards, e.g. a=1. Our new function is sin((x-1)/2)+1:
And finally, we want to stretch our function in the vertical axis, perhaps also by double, so b=2. (sin((x-1)/2)+1)*2: