This applet is really for teachers, because it shows my final thinking. See below for my thoughts on student use.

Working with students

With students, I would encourage an exploration of the reflections of the function and of each intercept and asymptote. Then they'll need a little algebraic thinking to refine the values of the parameters. Here's how I approached it - like a puzzle! It also helped to notice how similar the graphs were, so I knew the equations would be similar, too.
Quick questions:
1. How do you you find the value of x when y=0?
2. How do you make y=1 when x=0?
3. How do you make y=-2 when x gets infinitely big or small?
4. How do you verify that y gets infinitely big or small when x=-1.5?
Quick answers:
1. Set numerator to zero.
2. The constants must be equal.
3. The numerator's x-coefficient is twice the denominator's.
4. Set denominator to zero.