1.8: Stationary points and sketch of a rational curve

4PM1/2/November/2020 — Question 9 · 18 marks

A curve

\displaystyle C

has equation

y=\frac{2+4x-x^2}{2x+1}, \qquad x\neq-\frac12.

(a) Write the equation of

\displaystyle C

in the form

ax^2+(by-4)x+(y-c)=0,

where

\displaystyle a

\displaystyle b

and

\displaystyle c

are integers whose values are to be found.

(3)

(b) Hence show that

\displaystyle x

is real when

\displaystyle y\leq2

and when

\displaystyle y\geq3

(4)

\displaystyle C

(6)

(d) Sketch

\displaystyle C

, showing clearly

(i) the exact coordinates of the points where

\displaystyle C

crosses the

\displaystyle x

-axis,

(ii) the asymptote to

\displaystyle C

that is parallel to the

\displaystyle y

-axis,

(iii) the coordinates of the stationary points.

(5)