Repeat a square-root and line motif · source reference

Printed question

A curve has equation $y=f(x)$, where

$$f(x)=\begin{cases}\sqrt{8x+1}&\text{for }0\le x\le3,\\11-2x&\text{for }3<x\le5,\end{cases}$$

and $f(x)=f(x-5)$ for all real values of $x$.

(i) Sketch the curve for $-2\le x\le8$. [2]

(ii) On a separate diagram, sketch the curve with equation $y=f(\frac12x-1)$, for $-2\le x\le8$. [2]

[2021/CJC/Prelim/I/1]

Printed working

(i) Printed graph has points $(-2,5),(0,1),(3,5),(5,1),(8,5)$; curved increasing sections in blue, straight decreasing sections in red.

(ii) Printed transformed graph has points $(-2,5),(0,3),(2,1),(8,5)$.

$y=f(x)$ → translate 1 unit in the positive $x$-direction → $y=f(x-1)$ → scale parallel to $x$-axis by a scale factor of 2 → $y=f(\frac12x-1)$.

Teacher notes

$f(x)=f(x-5)$: Curve repeats itself after an interval of 5 units along the $x$-direction.

Effect: Increase $x$-coordinates of all points on graph of $y=f(x)$ by 1 unit, then multiply by 2.

Printed answer key

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

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.