How do changes to the coefficient of x affect the graph y = f(x) as opposed to changes to the coefficient of f(x)?

If you change the coefficient of x, it will stretch or squash the graph in the x direction by the reciprocal of the factor that the coefficient is changed by. If you change the coefficient of f(x), then the graph will be stretched or squashed in the y direction by the same factor that the coefficient is changed by. The graph is stretched or squashed about the origin.
E.g. if the graph y = f(x) has a maximum at (4,7), then the graph y = 2f(x) will have a maximum at (4,14), but the graph of y = f(2x) will have a maximum at (2,7).