Determine a sequence of transformations which maps the graph of $y = ext{cos} heta$ onto the graph of $y = 3 ext{cos} heta + 3 ext{sin} heta$ - AQA - A-Level Maths Mechanics - Question 5 - 2017 - Paper 2
Question 5
Determine a sequence of transformations which maps the graph of $y = ext{cos} heta$ onto the graph of $y = 3 ext{cos} heta + 3 ext{sin} heta$.
Fully justify y... show full transcript
Worked Solution & Example Answer:Determine a sequence of transformations which maps the graph of $y = ext{cos} heta$ onto the graph of $y = 3 ext{cos} heta + 3 ext{sin} heta$ - AQA - A-Level Maths Mechanics - Question 5 - 2017 - Paper 2
Step 1
Identify the transformation type
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Answer
To map the function from y=extcosheta to y=3extcosheta+3extsinheta, we need to interpret the desired transformation in terms of stretching and translating the function.
Step 2
Determine the amplitude and transformations
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Answer
The transformation involves expressing 3extcosheta+3extsinheta in the form Rextcos(heta−heta0), where:
Calculate the magnitude:
R=extsqrt(32+32)=extsqrt(18)=3extsqrt(2)
Find the angle:
heta_0 = an^{-1} rac{3}{3} = rac{ ext{π}}{4}
Step 3
State the transformations
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Answer
Stretch in the y-direction by a factor of 3extsqrt(2): This scales the amplitude of the cosine function.
Translate the graph: The function can be written as:
y = 3 ext{cos}( heta - rac{ ext{π}}{4})
Transformation includes a right translation of - rac{ ext{π}}{4}.
