Porisma

Worked solution

Halve horizontal distances

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

CJC · Transformations · Example 3

State the horizontal scaling

Answer

Scale parallel to the x-axis by factor 1/21/2.

Replace x with 2x in the original equation. Solve the point relation 2x=p2x=p to obtain (p,q)↦(p/2,q)(p,q)\mapsto(p/2,q); the scale factor is the reciprocal of the input multiplier.

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Sketch separately and transform the asymptotes

Answer

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

Preserve both branches on each separate diagram. The original vertical asymptote transforms as x=−1↦2x=−1  ⟹  x=−1/2x=-1\mapsto2x=-1\implies x=-1/2. The horizontal asymptote y1 is independent of x, so it stays unchanged. These source remarks continue on PDF 7 / printed 79. Independently, the y-intercept remains 2; the x-intercept moves from −2 to −1.

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Separate original 1+1/(x+1) and scaled 1+1/(2x+1), both branches. Vertical asymptote moves from −1 to −1/2; horizontal asymptote remains y=1.
Separate original 1+1/(x+1) and scaled 1+1/(2x+1), both branches. Vertical asymptote moves from −1 to −1/2; horizontal asymptote remains y=1.
Separate original 1+1/(x+1) and scaled 1+1/(2x+1), both branches. Vertical asymptote moves from −1 to −1/2; horizontal asymptote remains y=1.
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