Distinguish an absolute input from the order of affine transformations

(a) The supplied graph $y=f(x)$ has asymptotes y=0, y=2, x=−1, x=1, maximum (3,3), minimum at O and the right zero (2,0) shown. Sketch:

(i) $y=f(|1+x|)$;3 marks

(ii) $y=1/f(x)$.3 marks

(b) Alfred and Betty are to transform $y=f(x)$ to $y=5f(2x+1)-3$, with original $f(x)=x^2$ and target $y=5(2x+1)^2-3$.

Alfred’s steps, in order: 1.Scale parallel to y by 5; 2.Translate down 3; 3. Scale parallel to x by 1/2; 4. Translate along x by −1.

Betty’s steps, in order: 1.Translate along x by −1; 2. Scale parallel to x by 1/2; 3. Scale parallel to y by 5; 4. Translate down 3.

One person made an error in one step.

(i) Identify the person and incorrect step.2 marks

(ii) Give the final equation resulting from that incorrect sequence.1 mark