A-LevelFurther MathematicsRational functions and graphs9231/12 · Oct/Nov 2022 · Q715 Marks

The curve C has equation `y = (x²-x)/(x+1)`.

(a)Find the equations of the asymptotes of C.[3]
(b)Find the exact coordinates of the stationary points on C.[4]
(c)Sketch C, stating the coordinates of any intersections with the axes.[3]
(d)Sketch the curve with equation `y = |(x²-x)/(x+1)|` and find in exact form the set of values of x for which `|(x²-x)/(x+1)| < 6`.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[B1] `x=-1` (States vertical asymptote.)
[M1] Finds oblique asymptote. (Polynomial division `(x²-x)/(x+1) = x-2 + 2/(x+1)`.)
[A1] `y=x-2`
[M1 A1] Differentiates and sets derivative equal to 0. (`dy/dx = 1 - 2(x+1)⁻² = 0 implies (x+1)²=2 implies x = -1 ± √2`.)
[A1 A1] `(-1+√2, -3+2√2)`, `(-1-√2, -3-2√2)`
[B1] Axes and asymptotes labelled.
[B1] Upper branch with (0,0) and (1,0) stated or clear on scale.
[B1] Lower branch correct and all asymptotic approaches correct.
[B1 FT] FT from sketch in part (c).
[M2] Finds critical points. (For `(x²-x)/(x+1) = 6 implies x²-7x-6=0 implies x=(7 ± √73)/2`. For `(x²-x)/(x+1) = -6 implies x²+5x+6=0 implies x=-3, -2`. Award M1 for each case.)
[A1 A1] `-3 < x < -2` and `(7-√73)/2 < x < (7+√73)/2`

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