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Date November 2020 Marks available 5 Reference code 20N.1.AHL.TZ0.H_10
Level Additional Higher Level Paper Paper 1 (without calculator) Time zone Time zone 0
Command term Sketch Question number H_10 Adapted from N/A

Question

Consider the function fx=ax3+bx2+cx+d , where x and a, b, c, d

Consider the function gx=12x3-3x2+6x-8, where x.

The graph of y=g(x) may be obtained by transforming the graph of y=x3 using a sequence of three transformations.

Write down an expression for f'x.

[1]
a.i.

Hence, given that f1 does not exist, show that b23ac>0.

[3]
a.ii.

Show that g1 exists.

[2]
b.i.

g(x) can be written in the form p(x2)3+q , where p, q.

Find the value of p and the value of q.

[3]
b.ii.

Hence find g-1(x).

[3]
b.iii.

State each of the transformations in the order in which they are applied.

[3]
c.

Sketch the graphs of y=g(x) and y=g-1(x) on the same set of axes, indicating the points where each graph crosses the coordinate axes.

[5]
d.

Markscheme

f'x=3ax2+2bx+c        A1


[1 mark]

a.i.

since f1 does not exist, there must be two turning points       R1

(f'x=0 has more than one solution)

using the discriminant Δ>0        M1

4b2-12ac>0        A1

b2-3ac>0        AG


[4 marks]

a.ii.

METHOD 1

b2-3ac=-32-3×12×6        M1

=9-9

=0        A1

hence g1 exists        AG

 

METHOD 2

g'x=32x2-6x+6        M1

Δ=-62-4×32×6

Δ=36-36=0 there is (only) one point with gradient of 0 and this must be a point of inflexion (since gx is a cubic.)       R1

hence g1 exists        AG


[2 marks]

b.i.

p=12         A1

x-23=x3-6x2+12x-8          (M1)

12x3-6x2+12x-8=12x3-3x2+6x-4

gx=12x-23-4q=-4        A1


[3 marks]

b.ii.

x=12y-23-4          (M1)


Note: Interchanging x and y can be done at any stage.


2x+4=y-23          (M1)

2x+43=y-2

y=2x+43+2

g-1x=2x+43+2        A1


Note: g-1x= must be seen for the final A mark.


[3 marks]

b.iii.

translation through 20,          A1


Note: This can be seen anywhere.


EITHER
a stretch scale factor 12 parallel to the y-axis then a translation through 0-4          A2
OR
a translation through 0-8 then a stretch scale factor 12 parallel to the y-axis          A2


Note:
Accept ‘shift’ for translation, but do not accept ‘move’. Accept ‘scaling’ for ‘stretch’.


[3 marks]

c.

        A1A1A1M1A1


Note:
Award A1 for correct ‘shape’ of g (allow non-stationary point of inflexion)
Award A1 for each correct intercept of g
Award M1 for attempt to reflect their graph in y=x, A1 for completely correct g-1 including intercepts


[5 marks]

d.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.i.
[N/A]
b.ii.
[N/A]
b.iii.
[N/A]
c.
[N/A]
d.

Syllabus sections

Topic 2—Functions » SL 2.2—Functions, notation domain, range and inverse as reflection
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Topic 2—Functions » AHL 2.13—Rational functions
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