In function transformations, what does it mean when "f" has a coefficient? how does it affect the transformation? for example, 4f(x-2)+1.

For your example of      there is a vertical expansion by a factor of 4. all the y-coordinates get multiplied by 4.remember that expansions/compressions/reflections all come before translations.so there would be a vertical expansion by a factor of 4, and then translation 2 units right and 1 unit up.the y-coordinates of points on the original graph get multiplied by 4, and then they are translated.======in general: if there is  a coefficient outside, then there will be either a vertical expansion or vertical compression, and possible a reflection in the x-axis if negative.    if is greater than 1, there will be a vertical expansion by a factor of  .if   is between 0 and 1, there will be a vertical compression by a factor of  .if  is negative, then there will be a reflection in the x-axis.keep in mind that translations (shifts left/right and up/down) are always last before compressions, expansions, and reflections.