Absolute input discards the left-hand source features · source reference

Printed question

The diagram shows the graph of $y=f(x)$. The curve passes through the origin $O$, the points $A(-4a,0)$ and $C(-2a,0)$. The points $B(-3a,2)$ and $D(-a,-2)$ are stationary points and the lines $y=0$ and $x=a$ are asymptotes of the curve.

Sketch on separate clearly labelled diagrams, the graphs of

(i) $y=1-f(x)$; [3]

(ii) $y=|f(|x|)|$; [2]

(iii) $y=\frac1{f(x)}$, [3]

indicating clearly the points corresponding to $A,B,C,D$, and the asymptotes, where applicable.

[modified 2010/RVHS/Prelim/II/1]

Printed working

(i) $y=1-f(x)$.

$y=f(x)\xrightarrow{y\mapsto-y}y=-f(x)\xrightarrow{y\mapsto y-1}y=1-f(x)$.

Reflect about $x$-axis; translate by 1 unit in positive $y$-direction.

(ii) $y=|f(|x|)|$.

(iii) $y=\frac1{f(x)}$.

Teacher notes

(i) Effect: Change the sign of $y$-coordinates of all points on graph of $y=f(x)$, then increase the $y$-coordinates of all points by 1 unit.

(ii) 1. Sketch graph of $y=f(|x|)$ 1st: (a) For $x\ge0$, retain this part of graph of $y=f(x)$. (b) For $x<0$, discard this part of graph of $y=f(x)$ and replace it by a reflection of graph in (a) about the $y$-axis.

2. Then sketch the graph of $y=|f(|x|)|$ by (a) keeping the part of graph of $y=f(|x|)$ where $y\ge0$, (b) reflecting the part of graph of $y=f(|x|)$ where $y<0$ about the $x$-axis.

(iii) 1. Divide original graph into regions where $y>0,y<0$. Note $f(x)$ and $\frac1{f(x)}$ have the same sign.

2. Take the reciprocal of all $y$-coords on graph of $y=f(x)$ (i.e. $\frac1{f(x)}$).

3. Identify new intercepts, asymptotes and turning pts.

4. Within each region, complete the graph using the features identified in 2, 3.

Printed comparison table

$y=f(x)$	$y=1/f(x)$
Undefined at $x=a$ (i.e. vertical asymptote)	$x$-intercept $(a,0)$
Horizontal asymptote at $y=0$; as $x\to\infty,y\to0^-$	Undefined as $x\to\infty$; $y\to-\infty$
$x$-intercepts $(-4a,0),(-2a,0),(0,0)$; note $y=f(x)=0$	Vertical asymptotes $x=-4a,-2a,0$; note $1/f(x)$ undefined
Max pt $(-3a,2)$; min pt $(-a,-2)$	Min pt $(-3a,1/2)$; max pt $(-a,-1/2)$
As $x$ increases, $y$ increases; as $x$ increases, $y$ decreases	As $x$ increases, $y$ decreases; as $x$ increases, $y$ increases
$y>0$ (curve above $x$-axis); $y<0$ (curve below $x$-axis)	$y>0$ (curve above $x$-axis); $y<0$ (curve below $x$-axis)

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 (a,0) an x-intercept. Since f is undefined at a, its reciprocal is also undefined there: this is a limiting hole, not an attained intercept. The diagram is redrawn with an open point; the printed claim is retained above.

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.