Determine a sequence of transformations which maps the graph of
y =
cos θ
onto the graph of
y = 3cos θ + 3sin θ
Fully justify your answer. - AQA - A-Level Maths Pure - Question 5 - 2017 - Paper 2
Question 5
Determine a sequence of transformations which maps the graph of
y =
cos θ
onto the graph of
y = 3cos θ + 3sin θ
Fully justify your answer.
Worked Solution & Example Answer:Determine a sequence of transformations which maps the graph of
y =
cos θ
onto the graph of
y = 3cos θ + 3sin θ
Fully justify your answer. - AQA - A-Level Maths Pure - Question 5 - 2017 - Paper 2
Step 1
Identify the transformations needed to map $y = \cos \theta$ to $y = 3\cos \theta + 3\sin \theta$
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Answer
The graph can be rewritten in the form of Rcos(θ−α), where R is the amplitude and \alpha is the phase shift. Therefore, we compare:
y=3cosθ+3sinθ
with Rcos(θ−α).
This gives us:
Rcosα=3andRsinα=3.
Step 2
Calculate R and α
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Answer
To find R, we use:
R=(3)2+(3)2=18=32.
Next, we find \alpha:
\sin \alpha = \frac{3}{R} = \frac{3}{3\sqrt{2}} = \frac{1}{\sqrt{2}}$$.
This gives us \alpha = \frac{\pi}{4}$.
Step 3
Interpret the transformations
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Answer
The transformation involves:
Stretch in the y-direction by a factor of 32, hence:
y=32cos(θ).
Phase shift / translation to the right by \frac{\pi}{4}.
