A-LevelMathematicsFunctions9709/12 · May/June 2023 · Q812 Marks

The diagram shows the graph of y = f(x) where the function f is defined by f(x) = 3 + 2 sin(1/4 x) for 0 ≤ x ≤ 2π. [Figure 8.1]

(a)On the diagram above, sketch the graph of y = f⁻¹(x).[2]
(b)Find an expression for f⁻¹(x).[2]
(c)The diagram above shows part of the graph of the function g(x) = 3 + 2 sin(1/4 x) for −2π ≤ x ≤ 2π. Complete the sketch of the graph of g(x) on the diagram above and hence explain whether the function g has an inverse.[2]
(d)Describe fully a sequence of three transformations which can be combined to transform the graph of y = sin x for 0 ≤ x ≤ π/2 to the graph of y = f(x), making clear the order in which the transformations are applied.[6]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[*B1] The line y = x correctly drawn. Can be implied by reasonably correct graph of f⁻¹ (x).
[DB1] Fully correct (needs to reach y = 2π and x-axis and cross the line y = x in the correct squares).
[M1] y=3+2sin(1/4 x) leading to sin(1/4 x) = (y-3)/2 Attempting to arrive at an expression for sin(1/4 x); condone ± sign errors. Variables may be interchanged initially. M1 not implied by x = (y±3)/(2sin(1/4))
[A1] x=4sin⁻¹((y-3)/2) leading to [f⁻¹(x)or y =] 4sin⁻¹((x-3)/2) ISW Must clearly be sin⁻¹((x-3)/2) NOT sin⁻¹(x-3)/2 Allow (3-x)/(-2) but not (3-x)÷2
[B1] Continuing given graph from y intercept to -2π. The correct shape needed between 0 and -2π, including starting to level off (gradient in the final two squares needs to be reducing) as -2π is approached. The y co-ordinate at-2π must be in the correct square.
[B1] Yes it does have an inverse, because the graph is always increasing OR because it is one-one OR because it passes the horizontal line test OR it is not a many to one [function]. FT If there is no graph to the left of the y axis, no mark is available. FT an incorrect graph and if the answer is now ‘No’ provide an appropriate reason.
[B2,1,0] {Stretch} {factor 4} {in x-direction} B2 for fully correct, B1 with two elements correct. Condone use of 'sf' instead of factor and 'co-ordinates' stretched instead of graph stretched. Allow any mention of x-axis, horizontally or y-axis invariant. Wavelength or period increased by a factor of 4 for B2 or by 4 for B1.
[B2,1,0] {Stretch} {factor 2} {in y-direction} B2 for fully correct, B1 with two elements correct. Condone use of 'sf' instead of factor and 'co-ordinates' stretched instead of graph stretched. Allow any mention of y-axis, vertically or x-axis invariant. Allow y 'co-ordinates' doubled or amplitude doubled for B2.
[B2,1,0] {Translation} [0/3] B2 for fully correct, B1 with two elements correct. Allow shift. Any mention of y axis, y-direction or vertically implies {0}, so shift by 3 vertically is B2, but shift by a factor of 3 vertically or a translation of 3 'up' is B1.
[6] After scoring B2, B2 the final transformation can only be awarded B2 if the order is fully correct i.e. the translation must not be applied before the y stretch. If all correct except the order award B2B2B1.