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Date May 2022 Marks available 2 Reference code 22M.1.SL.TZ1.6
Level Standard Level Paper Paper 1 (without calculator) Time zone Time zone 1
Command term Describe Question number 6 Adapted from N/A

Question

Consider f(x)=4sinx+2.5 and g(x)=4sin(x-3π2)+2.5+q, where x and q>0.

The graph of g is obtained by two transformations of the graph of f.

Describe these two transformations.

[2]
a.

The y-intercept of the graph of g is at (0, r).

Given that g(x)7, find the smallest value of r.

[5]
b.

Markscheme

translation (shift) by 3π2 to the right/positive horizontal direction         A1

translation (shift) by q upwards/positive vertical direction         A1

 

Note: accept translation by (3π2q)

Do not accept ‘move’ for translation/shift.

 

[2 marks]

a.

METHOD 1

minimum of 4sin(x-3π2) is -4  (may be seen in sketch)          (M1)

-4+2.5+q7

q8.5  (accept q=8.5)         A1

substituting x=0 and their q (=8.5) to find r          (M1)

(r=)  4sin(-3π2)+2.5+8.5

4+2.5+8.5          (A1)

smallest value of r is 15         A1

 

METHOD 2

substituting x=0 to find an expression (for r) in terms of q          (M1)

(g(0)=r=)  4sin(-3π2)+2.5+q

(r=)  6.5+q         A1

minimum of 4sin(x-3π2) is -4          (M1)

-4+2.5+q7

-4+2.5+(r-6.5)7  (accept =)          (A1)

smallest value of r is 15         A1

 

METHOD 3

4sin(x-3π2)+2.5+q=4cosx+2.5+q         A1

y-intercept of 4cosx+2.5+q is a maximum          (M1)

amplitude of g(x) is 4          (A1)

attempt to find least maximum          (M1)

r=2×4+7

smallest value of r is 15         A1

 

[5 marks]

b.

Examiners report

Candidates knew aspects of the transformations performed but some were unable to correctly describe them fully, e.g., omitting direction (right/up/positive) or using 'move' instead of translate/shift. Each description requires three parts: transformation type, size and direction. e.g., translation of q units up. For part (b) few candidates were able to fully navigate the reasoning required in this question. A common error was to evaluate sin(-3π2)=-1, instead of 1. Those who used sketches to assist in their thinking were typically more successful.

a.
[N/A]
b.

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.7—Circular functions: graphs, composites, transformations
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Topic 3— Geometry and trigonometry

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