Practice section 8 of 9

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.

The given graph is $y=f(2x)$, passing through A=(−1,0), B=(0,2), C=(1,0), D=(3,−3). The diagram shows a right-hand horizontal asymptote y=−1.

Sketch on separate clearly labelled diagrams:

(i) $y=f(-2x)$; [1]

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

(iii) $y=\frac12f(2x)$. [2]

The given graph is $y=f(x)$, with maximum A=(2,2), B=(4,0), C=(0,1) and vertical asymptote x=5; its left tail approaches y=0. Sketch on separate diagrams:

(i) $y=f(2x-1)$; [3]

(ii) $y=f(x/2)-2$; [3]

(iii) $y=1/f(x)$; [4]

Show the corresponding A,B,C coordinates and all asymptotes where applicable.

The given graph $y=f(x)$ intersects the axes at A=(0,−2), B=(−1,0), has minimum C=(4,0.5), and asymptotes x=1, y=3.

Sketch separately:

(i) $y=f(-1-x)$; [4]

(ii) $y=1/f(x)$; [4]

Clearly indicate asymptotes, axial intercepts and corresponding A,B,C where appropriate.

(a) $y=x^2-1$ is translated 3 units in the positive x-direction, stretched by factor 2 parallel to y, then translated 2 units in the negative y-direction. Find the new equation as $y=f(x)$. [3]

(b) Three successive transformations change $y=h(x)$ to $y=h(1-2x)$. Describe them. [3]