Sketch, on separate diagrams, the curves \[\text{(i)}\ y=\frac{x-a}{x},\qquad \text{(ii)}\ y=\frac{x^2-a^2}{x^2},\] where \(a\) is a positive constant.

[The equations of any asymptotes should be stated, together with the coordinates of any intersections with the axes.]

When sketching curves, it is helpful to ask:

  1. Where are the intersections with the axes?

  2. What happens as \(x\) tends to \(\pm \infty\)? Are there horizontal asymptotes? Asymptotes at an angle?

  3. What are the vertical asymptotes?

  4. Are there any stationary points?

  5. Is there any symmetry?

Can we write our equation for (i) as \(y=p+\frac{q}{x}\) for some \(p\) and \(q\)?

Can we view this as the combination of two transformations of a simpler curve?

How is our equation for (ii) similar to that for (i)? How is it different?

Hence sketch the curves \[\text{(iii)}\ y = \left|\frac{x-a}{x}\right|,\qquad \text{(iv)}\ y^2 = \frac{x^2-a^2}{x^2}.\]

Will these functions look similar to the ones from the first part at all?