Porisma

Worked solution

Use zeros, poles and turning values to sketch three reciprocals

Given the three supplied graphs of y=f(x)y=f(x), sketch y=1/f(x)y=1/f(x) on separate diagrams.

(a) Zeros x=1,2, y-intercept 2, minimum (3/2,−1/4).

(b) Vertical asymptote x=−1, horizontal asymptote y=−1, y-intercept 1, zero x=1.

(c) Vertical asymptote x=−1, zeros x=±5/2, y-intercept 2, and the oblique guide shown through (0,1), (2,0).

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Three qualitative source graph reconstructions, not asserted exact source formulas. (a) zeros 1,2,y intercept 2, minimum (3/2,-1/4). (b) pole -1, horizontal -1,y intercept 1, zero 1. (c)[2009/VJC/I/7] pole -1, zeros ±5/2,y intercept 2, and the shown oblique guide through (0,1),(2,0). All labelled source features are retained.
Three qualitative source graph reconstructions, not asserted exact source formulas. (a) zeros 1,2,y intercept 2, minimum (3/2,-1/4). (b) pole -1, horizontal -1,y intercept 1, zero 1. (c)[2009/VJC/I/7] pole -1, zeros ±5/2,y intercept 2, and the shown oblique guide through (0,1),(2,0). All labelled source features are retained.
Three qualitative source graph reconstructions, not asserted exact source formulas. (a) zeros 1,2,y intercept 2, minimum (3/2,-1/4). (b) pole -1, horizontal -1,y intercept 1, zero 1. (c)[2009/VJC/I/7] pole -1, zeros ±5/2,y intercept 2, and the shown oblique guide through (0,1),(2,0). All labelled source features are retained.

Part (c): [2009/VJC/I/7]

CJC · Discussion · Question 6

Answer

(a) Poles 1,2, horizontal 0, y-intercept 1/2, maximum (3/2,−4). (b) Pole 1, horizontal −1, y-intercept 1, open (−1,0). (c) Poles ±5/2, horizontal 0, y-intercept 1/2, open (−1,0).

(a) Map the parabola-like source landmarks

The zeros 1,2 give reciprocal vertical asymptotes there. The y-intercept becomes (0,1/2). The minimum (3/2,−1/4) becomes a maximum (3/2,−4) on the negative branch between the poles. Since the original grows without bound at both ends, the reciprocal has horizontal asymptote y=0. It has no x-intercept.

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(b) Keep the original excluded input

The zero at 1 becomes a vertical asymptote x=1. The horizontal asymptote stays y=−1 and the y-intercept stays (0,1). At the original pole x=−1, the reciprocal tends to 0 but remains undefined: draw an open point (−1,0), not the printed filled point. There is no true x-intercept.

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(c) Reciprocal of a curve with an oblique asymptote

The zeros ±5/2 give reciprocal poles x=±5/2. The y-intercept becomes (0,1/2). Original unbounded tails give horizontal asymptote y=0. The original pole input −1 remains excluded, with an open limiting point (−1,0) rather than the source’s filled crossing. Preserve the source branch signs: positive left of −5/2, negative between −5/2 and −1, positive from −1 to 5/2, and negative after 5/2.

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Compare your sketch

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(a) reciprocal poles 1,2, horizontal 0,y intercept 1/2, maximum (3/2,-4) between poles. (b) pole 1, horizontal -1,y intercept 1, and open zero-limit point (-1,0). (c)poles ±5/2, horizontal 0,y intercept 1/2, and open zero-limit point (-1,0). Source filled points at excluded original pole inputs are qualified and drawn open; no actual x intercepts.
(a) reciprocal poles 1,2, horizontal 0,y intercept 1/2, maximum (3/2,-4) between poles. (b) pole 1, horizontal -1,y intercept 1, and open zero-limit point (-1,0). (c)poles ±5/2, horizontal 0,y intercept 1/2, and open zero-limit point (-1,0). Source filled points at excluded original pole inputs are qualified and drawn open; no actual x intercepts.
(a) reciprocal poles 1,2, horizontal 0,y intercept 1/2, maximum (3/2,-4) between poles. (b) pole 1, horizontal -1,y intercept 1, and open zero-limit point (-1,0). (c)poles ±5/2, horizontal 0,y intercept 1/2, and open zero-limit point (-1,0). Source filled points at excluded original pole inputs are qualified and drawn open; no actual x intercepts.

Common mistakes

The source sketches in(b), (c)fill a zero-limit point at x=−1. A limit is not a defined function value.

Solution notes

All three supplied graphs and all three reciprocal solution graphs reconstructed. Source graph-only data are represented by qualitative interpolations, not claimed exact formulas. The reciprocal domain holes are explicit qualifications. All labelled source features retained.

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