Reconstruct signs from two absolute-value sketches · source reference

Printed question

(a) Describe a sequence of exactly three transformations that will transform the curve with equation $y=\frac1{x-3}$ onto the curve with equation $y=\frac{2x+a}{3-x}$, where $a$ is a positive constant. [3]

Given instead that $a<-6$, use a non-graphical method to determine the range of values of $x$ where the graph of $y=\frac{2x+a}{3-x}$ is concave upwards. [2]

(b) The diagrams below show the graphs of $y=f(|x|)$ and $y=|f(x)|$, where the equations of the asymptotes and the coordinates of the turning points are given. The gradient of the graph with equation $y=f(x)$ at the origin is $-\frac34$.

On separate diagrams, sketch the graph of $y=f(x)$ including the coordinates of the points where the graphs cross the $x$- and $y$-axes and the equations of the asymptotes, if any. [2]

[2018/NJC/Prelim/I/8]

Printed working

(a) $y=\frac{2x+a}{3-x}=-2-\frac{6+a}{x-3}$.

1. Reflect graph about the $x$-axis.
2. Scale graph parallel to the $y$-axis by factor of $(6+a)$.
3. Translate graph in the negative $y$-direction by 2 units.

$y=\frac1{x-3}\xrightarrow{y\mapsto-y}y=-\frac1{x-3}\xrightarrow{y\mapsto y/(6+a)}y=\frac{6+a}{3-x}\xrightarrow{y\mapsto y+2}y=-2+\frac{6+a}{3-x}$.

$y=\frac{2x+a}{3-x}=-2-\frac{6+a}{x-3}$.

$\frac{dy}{dx}=-\frac{6+a}{(x-3)^2}(-1)=\frac{6+a}{(x-3)^2}$.

$\frac{d^2y}{dx^2}=\frac{6+a}{(x-3)^3}(-2)=-\frac{2(6+a)}{(x-3)^3}$.

Where the graph is concave upwards, $\frac{d^2y}{dx^2}>0$.

Since $a<-6,6+a<0$. Therefore, $(x-3)^3>0\Rightarrow x>3$.

(b) Native graph below preserves the printed shape and labels.

Teacher notes

(b) 1. For $x\ge0$, this part of the graph of $y=f(x)$ is the same as the graph of $y=f(|x|)$.

2. For $x<0$, since the gradient of curve $y=f(x)$ at the origin is negative, this part of the graph of $y=f(x)$ is the same as the graph of $y=|f(x)|$.

Printed answer key

(a) 1. Reflect graph about x-axis. 2. Scale parallel to y-axis by factor (6+a). 3. Translate negative y-direction by 2 units; x>3. (b) The native graph 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.