Discussion questions

Sketch each graph, clearly indicating asymptotes, axial intercepts and coordinates of any turning points.

(a) $y=2x^3-5x^2+x-2$.

(b) $y=4x^2e^x$.

(c) $y=\ln x/(2x)$.

(d) $y=(x^2+2x-8)/(x^2+2x-3)$.

(e) $y=|2x-x^2|$.

(f) $y=(ax+1)/(x-b)$, where (i)−1<a<0,0<b<1; or (ii)a<−1,b>1. What happens if ab+1=0?

C has equation $y=x^2/(x-2)$.

(i) Find the asymptotes.1 mark

(ii) Sketch C, labelling asymptotes, axial intercepts and turning points.2 marks

(iii) Hence find the values of k for which $x^2=k(x^2-4)$ has no real roots.2 marks

C has equation $y=(x^2+9x+16)/(x+2)$.

(i) Find its asymptotes.

(ii) Find, in exact form, all k for which $k=(x^2+9x+16)/(x+2)$ has no real roots.

(iii) Sketch C with asymptotes, stationary points and x-/y-axis intersections labelled.

$C_1$ has equation $y=(ax^2+bx)/(x+c)$.

(i) It has vertical asymptote x=2 and an oblique asymptote through (0,10) and (−2,0). Find a,b,c.

(ii) Hence sketch $C_1$, clearly stating all asymptotes and stationary points.

C has equation $y=(x^2+ax+4)/(x+b)$, with vertical asymptote x=−1 and a stationary point at x=2.

(i) Find a and b.

(ii) Find the other asymptote.

(iii) Without a calculator, find the set of values y cannot take.

(iv) Sketch C, showing axial intercepts, asymptotes and stationary points.

(v) Deduce the number of real roots of $(4-x^2)(x+1)^2=(x^2-4x+4)^2$.

· Techniques of Differentiation

C has equation $y=(2x^2+ax+3a)/(x+b)$, where a,b are nonzero constants. x=2 is an asymptote.

(i) State b.

(ii) By differentiation, find the range of a for which C has no stationary points.

(iii) Given a=−3, sketch C with all asymptotes and axis intersections.