Discussion questions · Section 2 of 2

Curve C is $f(x)=(x-9)^2/[(x-3)(x+3)]$.

(i) Express it as $P+Q/(x-3)+R/(x+3)$.

(ii) State all asymptotes and axial intercepts.

(iii) Sketch separately (a) $y=f(x)$ and (b) $y=|1/f(x)|$, making the main features clear.

The given graph is $G(x)=f(3-x/2)$, with stationary points (8,−2), (14,2) and x-intercepts 6,10,18. The source diagram shows a vertical asymptote x=0 and left horizontal asymptote y=−4.

Sketch separately:

(a) $y=1/f(3-x/2)=1/G(x)$;

(b) $y=f(x)$.

Clearly show asymptotes, stationary points and axial intercepts.

(i) State transformations taking $y=x^2$ to $y=\frac14(x-3)^2$.

Define

$$f(x)=\begin{cases}1,&0\le x\le1,\\ \frac14(x-3)^2,&1<x\le3,\\0,&\text{otherwise}.\end{cases}$$

(ii) Sketch f for −1≤x≤4.

(iii) Separately sketch $y=1+f(x/2)$ for −1≤x≤4.