Porisma

Worked solution

Move a reciprocal asymptote to the right

State the transformation which transforms y=1/xy=1/x to y=1/(x−1)y=1/(x-1). On separate diagrams, sketch the two graphs, stating the equations of any asymptotes.

CJC · Transformations · Example 1

State the transformation

Answer

Translate 1 unit in the positive x-direction.

Replace xx by x−1x-1 in y=1/xy=1/x to obtain y=1/(x−1)y=1/(x-1). The point mapping is (p,q)↦(p+1,q)(p,q)\mapsto(p+1,q).

How did this part go?

Sketch separately and state the asymptotes

Answer

Original: x=0x=0, y=0y=0. Translated: x=1x=1, y=0y=0.

Draw both reciprocal branches on each diagram. The original branches lie in quadrantsI/III. The translated right branch lies above y0 for x>1; its left branch lies below y0 for x<1 and crosses the y-axis at −1 (an independent coordinate check). Apply the replacement to the old vertical asymptote: x=0↦x−1=0  ⟹  x=1x=0\mapsto x-1=0\implies x=1. The horizontal asymptote y=0y=0 is independent of x and remains unchanged. These remarks continue on PDF 3 / printed 75.

Expand diagram

Use the scrollbars or arrow keys to inspect details.

Separate graphs of 1/x with asymptotes x=0, y=0, and 1/(x−1) with asymptotes x=1, y=0. Both branches retained; translation 1 right.
Separate graphs of 1/x with asymptotes x=0, y=0, and 1/(x−1) with asymptotes x=1, y=0. Both branches retained; translation 1 right.
Separate graphs of 1/x with asymptotes x=0, y=0, and 1/(x−1) with asymptotes x=1, y=0. Both branches retained; translation 1 right.
How did this part go?