Review practice · Section 2 of 2

A translates 2 units in the negative x-direction; B reflects in the x-axis; C scales parallel to y by 2 with the x-axis invariant. Applied in order A,B,C, the resulting equation is $y=-2(x+3)/(x+2)$. Find the equation before the transformations.3 marks

(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

The supplied graphs are $y=f(|x|)$ and $y=|f(x)|$. Both have maximum (4, a) and vertical tangents x=0, x=6. The first is even, with maxima (−4, a), (4, a), zeros (−6,0), (0,0), (6,0), and slant asymptotes y=x+2 on the left and y=−x+2 on the right. The second is nonnegative, with zeros (0,0), (6,0), and slant asymptotes y=−x+2 on the left and y=x−2 on the right.

Given f itself has only one oblique asymptote, sketch $y=f(x)$, stating asymptotes, turning points and axial intercepts wherever possible.2 marks