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Original question & working

Reflect then reciprocate; translate then fold · source reference

modified 2010/ACJC/Prelim/I/3 · Question pp. 2 · Working pp. 6

Open the interactive lesson

The working and teaching notes are included in the interactive lesson.

Source solution PDF p.6 labels the reciprocal limit (2,0) as an intercept, although the original input remains excluded; the authored version marks a hole unless a continuous extension is intended. The same page labels the absolute-output image of (−1,0) as (−1,−4); its correct height is |0−4|=4.

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Full original question, working & notes

Printed question

The diagram below shows the graph of y=f(x)y=f(x). The curve cuts the axes at (−1,0)(-1,0) and (0,3)(0,3). It has a maximum point at (3,5)(3,5), and asymptotes x=−2x=-2 and y=2y=2.

Source graph with maximum (3,5) and asymptotes

Sketch on separate diagrams, the graphs of

(i) y=1f(−x)y=\frac1{f(-x)}; [3]

(ii) y=∣f(x)−4∣y=|f(x)-4|. [3]

showing clearly asymptotes, intercepts and coordinates of turning points, where available.

[modified 2010/ACJC/Prelim/I/3]

Printed working

(i) y=f(x)→sketch reciprocal graphy=1f(x)→x↦−xy=1f(−x)y=f(x)\xrightarrow{\text{sketch reciprocal graph}}y=\frac1{f(x)}\xrightarrow{x\mapsto-x}y=\frac1{f(-x)} (Reflect about yy-axis; OR vice versa).

(ii) y=f(x)→y↦y+4y=f(x)−4→put in modulus signy=∣f(x)−4∣y=f(x)\xrightarrow{y\mapsto y+4}y=f(x)-4\xrightarrow{\text{put in modulus sign}}y=|f(x)-4|.

Translate by 4 units in negative yy-direction; reflect the part of curve where y<0y<0 above xx-axis.

Native source reconstruction, question 4 diagram i Native source reconstruction, question 4 diagram ii

Printed comparison table

y=f(x)y=f(x)y=1/f(x)y=1/f(x)
Vertical asymptote x=−2x=-2xx-intercept (−2,0)(-2,0)
Horizontal asymptote y=2y=2Horizontal asymptote y=1/2y=1/2
xx-intercept (−1,0)(-1,0)Vertical asymptote x=−1x=-1
Max pt (3,5)(3,5)Min pt (3,1/5)(3,1/5)

Printed answer key

The answer key repeats the native diagrams above with the same coordinates and asymptote labels; no additional algebraic answer is printed.

Editorial qualification

The printed reciprocal table calls (-2,0) an x-intercept. It is a limiting hole because f is undefined at -2. After reflection the hole is (2,0). The source transformed modulus sketch labels (-1,-4), but a modulus output cannot be negative: its correct image is (-1,4). Printed labels are preserved in the native diagram as source labels, with this qualification.

Source provenance

2025 TJC H2 Math (9758) JC2 MSM: GRAPHS AND CURVE SKETCHING. Question/answer booklet printed pages 1–14; solutions printed pages 1–28. Running metadata and repeated headings are transcribed in the topic metadata article.