I'm going to assume that you're discussing the linear function y = mx + b. If that's not it, please provide more information on the function you're studying.
y = mx + b has a graph that is the straight line through (0,b) with slope m.
y = m(x-5) + b has a graph which is parallel to the previous graph, but intersects the x-axis at a different point.
Examples: start with y = mx + b
Suppose we have y = 3x + 2 to work with
The graph intersects the y-axis at (0,2). It intersects the x-axis where y=0=3x+2, or at x = -2/3 => (-2/3, 0)
Now modify y = 3x + 2 by replacing "x" with "x-5." the graph of this "new" function f(x) = 3(x-5) + 2 is parallel to the original graph, but lies 5 units to the right of the original graph. Both lines have slope 3.
In this case, if x increases by 4 (as your problem states), y will increase by 3(4), or 12, units.
If we go back to y = m(x-5), we see that the graph of this function intersects the x-axis at x=5, and this graph has the (unknown) slope m.
If x increases by 4, y increases by (m)(4), or 4m.