Porisma

Worked solution

Reciprocate a sketch with an excluded vertical limit

The curve shown has equation y=f(x)y=f(x). On a separate diagram, sketch y=1/f(x)y=1/f(x), indicating clearly the asymptotes and the coordinates of points corresponding to A(3,−1)A(3,-1) and B(1,0)B(1,0) where applicable.

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Given schematic y=f(x): left horizontal asymptote y=2, divergence upwards on both sides of x=0, root B(1,0), minimum A(3,−1), and right branch approaching y=0 from below. No formula is supplied.
Given schematic y=f(x): left horizontal asymptote y=2, divergence upwards on both sides of x=0, root B(1,0), minimum A(3,−1), and right branch approaching y=0 from below. No formula is supplied.
Given schematic y=f(x): left horizontal asymptote y=2, divergence upwards on both sides of x=0, root B(1,0), minimum A(3,−1), and right branch approaching y=0 from below. No formula is supplied.

CJC · Transformations · Example 18

Map the labelled points and limits

Answer

A↦A′(3,−1)A\mapsto A\prime(3,-1); BB produces vertical asymptote x=1x=1; left horizontal asymptote y=1/2y=1/2.

The given curve tends to 2 at the left infinite end, diverges at x=0x=0, crosses at B and has minimum A before tending to 0 from below at the right infinite end. A is unchanged because 1/(−1)=−11/(-1)=-1, but becomes a reciprocal maximum. B has no finite corresponding point because division by 0 is undefined. The left limit 2 becomes 1/2; the right zero limit makes the reciprocal unbounded, rather than creating a right horizontal asymptote.

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Sketch with the original domain exclusion

Answer

The origin is an open hole; the only indicated vertical asymptote is x=1x=1.

The Teacher fills the blank Student answer axes, labelling y=1/2y=1/2, x=1x=1 and A′(3,−1)A\prime(3,-1). Both original sides diverge upwards near 0; the reciprocal tends to 0 from above on both sides. Preserve the exclusion at 0. The source draws a filled origin, following its table convention; this is qualified as a hole without an explicitly defined extension. Preserve the negative branch to the right of 1 and its maximum at A′. No unique formula or unknown intercept is inferred.

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Given curve has left horizontal limit 2, vertical divergence at 0, root B(1,0), minimum A(3,−1), right limit 0 from below. Reciprocal has left horizontal asymptote 1/2, pole 1, open hole (0,0), unchanged A′(3,−1) as a maximum, and unbounded negative right tail.
Given curve has left horizontal limit 2, vertical divergence at 0, root B(1,0), minimum A(3,−1), right limit 0 from below. Reciprocal has left horizontal asymptote 1/2, pole 1, open hole (0,0), unchanged A′(3,−1) as a maximum, and unbounded negative right tail.
Given curve has left horizontal limit 2, vertical divergence at 0, root B(1,0), minimum A(3,−1), right limit 0 from below. Reciprocal has left horizontal asymptote 1/2, pole 1, open hole (0,0), unchanged A′(3,−1) as a maximum, and unbounded negative right tail.
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