Classifying the Reflection and Stretching of the Reciprocal Function

We present a simple criterion to determine if the graph of a rational function of the form $ f(x)=\frac{ax+b}{cx+d} $ includes a reflection of the reciprocal function $ r(x)=\frac{1}{x} $ over the $ x $-axis. In addition, we determine an algebraic expression for the factor by which the graph of $ y=r(x) $ is stretched vertically to produce the graph of $ y=f(x) $.