Porisma

Worked solution

Double the horizontal asymptote height

State the transformation which transforms y=1+1/(x+1)y=1+1/(x+1) to y=2(1+1/(x+1))y=2(1+1/(x+1)). On separate diagrams, sketch the two graphs, stating the equations of any asymptotes.

CJC · Transformations · Example 4

State the vertical scaling

Answer

Scale parallel to the y-axis by factor 2.

Replace y by y/2: y/2=1+1/(x+1)y/2=1+1/(x+1), so y=2(1+1/(x+1))y=2(1+1/(x+1)). The Teacher fills both the scale-factor statement and the intermediate y/2 equation. The point mapping is (p,q)↦(p,2q)(p,q)\mapsto(p,2q).

How did this part go?

Sketch separately and transform the asymptotes

Answer

Original: x=−1x=-1, y=1y=1. New: x=−1x=-1, y=2y=2.

Both original and transformed graphs have two branches. The source sketches and remarks continue on PDF 9 / printed 81. Apply the replacement to the old horizontal asymptote: y=1↦y/2=1  ⟹  y=2y=1\mapsto y/2=1\implies y=2. The vertical asymptote x−1 is independent of y and stays fixed. Independently, the y-intercept doubles from 2 to 4; the x-intercept stays −2.

Expand diagram

Use the scrollbars or arrow keys to inspect details.

Separate original 1+1/(x+1) and doubled 2(1+1/(x+1)), both branches. Horizontal asymptote changes from y=1 to y=2; vertical asymptote remains x=−1.
Separate original 1+1/(x+1) and doubled 2(1+1/(x+1)), both branches. Horizontal asymptote changes from y=1 to y=2; vertical asymptote remains x=−1.
Separate original 1+1/(x+1) and doubled 2(1+1/(x+1)), both branches. Horizontal asymptote changes from y=1 to y=2; vertical asymptote remains x=−1.
How did this part go?