Suppose X has density f(x), and we make a linear transformation Y=aX+b. What is the distribution of Y? This can be computed easily using the Change of Variables formula. The constant a changes the scale of measurement for X, while the constant b shifts the location of X. Taking together all possible values of a and b creates a location-scale family of densities, since it allows for all possible locations and scales for the density of Y.
SOLUTION: Y=g(X)=aX+b. First we must compute the INVERSE function of g by solving for Y in terms of X. This is easily done to get: X=(Y-b)/a=h(Y). Second, Note that h'(y)=1/a. So we can compute the differential element dx=h'(y)dy = (1/a)dy. Third Substitution into the density of X gives us the density of Y -- let us call it d(y) -- d(y)=f(h(y))h'(y)=f((y-b)/a)(1/a). Fourth: Compute the range of possible values of Y. If X take values on the range [xlo,xhi] then Y takes values on the range [(a*xlo+b),(a*xhi+b)]. This completes the calculation of the density of Y.
The set of all possible densities of Y, when a,b take all possible values, is called a location-scale family of densities. Note that a cannot take the value 0, because then the transformation from X to Y is not one to one; all values of X gets mapped into a single value of Y=b.