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Sketch the graph of y = cot t( x - π/2 ) for 0 ≤ x ≤ 2π - AQA - A-Level Maths Pure - Question 7 - 2022 - Paper 1
Question 7
Sketch the graph of y = cot t( x - π/2 ) for 0 ≤ x ≤ 2π
Worked Solution & Example Answer:Sketch the graph of y = cot t( x - π/2 ) for 0 ≤ x ≤ 2π - AQA - A-Level Maths Pure - Question 7 - 2022 - Paper 1
Step 1
Sketch the graph of y = cot( x - π/2 )
Answer
To sketch the graph of the function, we first need to analyze its behavior:
Step 1: Determine the Asymptotes
The cotangent function, cot(x), has vertical asymptotes where x = nπ for n being an integer. For our function, shifting the cotangent left by π/2 gives:
- Asymptotes at x = π/2 and x = 3π/2.
Step 2: Identify the Key Points
The cotangent function has specific behavior around its asymptotes:
- Approaches infinity as it approaches the asymptote from the left.
- Crosses the x-axis halfway between the asymptotes at x = π/2 and x = 3π/2, which is at x = π.
Step 3: Plot the Function
- From x = 0 to x = π/2, the function decreases from positive infinity to zero.
- From x = π/2 to x = 3π/2, it moves from zero back to negative infinity.
- Finally, from x = 3π/2 to 2π, the graph again decreases from negative infinity to zero.
Final Sketch
Thus, the graph consists of three branches:
- The first branch (0 to π/2) approaching the vertical asymptote.
- The second branch (π/2 to 3π/2) crossing at x = π.
- The third branch (3π/2 to 2π) approaching the next vertical asymptote.
No asymptotes need to be drawn on the final sketch, but ensure to represent the behavior correctly.