Map points through three graph transformations · source reference

Printed question

The diagram above shows the graph of $y=f(x)$.

On separate diagrams, sketch the graphs of

(i) $y=2f(x)+1$, [3]

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

(iii) $y=f(1-2x)$. [3]

showing in each case, the coordinates of the points corresponding to $A,B,C$ and the equations of the asymptotes.

[2010/SRJC/Prelim/II/3]

Printed working

(i) $y=2f(x)+1$.

$y=f(x)\xrightarrow{y\text{ replaced by }\frac12y}y=2f(x)\xrightarrow{y\text{ replaced by }y-1}y=2f(x)+1$.

Scale parallel to $y$-axis by factor of 2; translate by 1 unit in positive $y$-direction.

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

(iii) $y=f(1-2x)$.

$y=f(x)\xrightarrow{x\mapsto x+1}y=f(x+1)\xrightarrow{x\mapsto2x}y=f(2x+1)\xrightarrow{x\mapsto-x}y=f(-2x+1)$.

Translate 1 unit in negative $x$-direction; scale parallel to $x$-axis by factor $\frac12$; reflect about the $y$-axis.

Teacher notes

(i) Effect: Multiply $y$-coordinates of all points on graph of $y=f(x)$ by 2, then increase (i.e. add to) the resulting value by 1 unit.

(ii) Sketch of graph of $y=f(|x|)$:

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

(iii) Effect: Decrease $x$-coordinates of all points on graph of $y=f(x)$ by 1 unit, then multiply by $\frac12$, and finally change their sign.

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 abbreviated answer-key diagram for (iii) labels the point at the y-axis as B′(0,1), whereas the full worked diagram labels B′(0,−1). Since f(1)=−1, the correct value is −1; the key’s positive sign is preserved here as a printed variant and the native worked graph uses −1.

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.