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8 (a) Determine a sequence of transformations which maps the graph of $y = ext{sin} x$ onto the graph of $y = rac{ ext{√}3}{2} ext{sin} x - 3 ext{cos} x + 4$ - AQA - A-Level Maths Pure - Question 8 - 2018 - Paper 2
Question 8
8 (a) Determine a sequence of transformations which maps the graph of $y = ext{sin} x$ onto the graph of $y = rac{ ext{√}3}{2} ext{sin} x - 3 ext{cos} x + 4$. F... show full transcript
Worked Solution & Example Answer:8 (a) Determine a sequence of transformations which maps the graph of $y = ext{sin} x$ onto the graph of $y = rac{ ext{√}3}{2} ext{sin} x - 3 ext{cos} x + 4$ - AQA - A-Level Maths Pure - Question 8 - 2018 - Paper 2
Step 1
Determine a sequence of transformations which maps the graph of $y = sin x$ onto the graph of $y = \frac{\sqrt{3}}{2} \sin x - 3 \cos x + 4$
Answer
To determine the transformations, we compare the given functions:
- The original function is (y = \text{sin} x).
- The transformed function can be expressed as (y = R \sin(x + \alpha) + d), where (R = \sqrt{(\frac{\sqrt{3}}{2})^2 + (-3)^2} = \sqrt{\frac{3}{4} + 9} = \sqrt{\frac{39}{4}} = \frac{\sqrt{39}}{2}).
- The angle (\alpha) is determined from (R \cos \alpha = \frac{\sqrt{3}}{2}) and (R \sin \alpha = -3). Hence, (\tan \alpha = \frac{-3}{\frac{\sqrt{3}}{2}} = -\frac{6}{\sqrt{3}}).
- The transformations include:
- A vertical stretch by a factor of (\frac{\sqrt{39}}{2}) (scale in the y-direction).
- A translation up by 4 units.
- A translation left by the angle (\alpha).
Step 2
Show that the least value of \( \frac{1}{\text{√}3 \text{sin} x - 3 \text{cos} x + 4} \) is \( \frac{2 - \text{√}3}{2} \)
Answer
We need to analyze the function (f(x) = \text{√}3 \text{sin} x - 3 \text{cos} x + 4).
- To find the least value, we first find the critical points:
- The least value occurs when (\sin(x + \frac{\pi}{3}) = -1), so we compute:
- The minimum value is (\text{√}3(-1) - 3(0) + 4 = -\text{√}3 + 4 = 4 - \text{√}3).
- Thus, the least value of (\frac{1}{f(x)}) is derived from:
- (\frac{1}{4 - \text{√}3}) which simplifies to the required expression.
Step 3
Find the greatest value of \( \frac{1}{\text{√}3 \text{sin} x - 3 \text{cos} x + 4} \)
Answer
- The greatest value occurs when (\text{√}3 \text{sin} x - 3 \text{cos} x + 4) is minimized, leading to:
- The maximum value can be calculated or inspected using the earlier minimum found.
- So, we evaluate the parameter substitution analogous to the previous part leading us to derive the final answer as (\frac{2 + \text{√}3}{2}).