Sketch the graph of y = cot t( x - π/2 ) for 0 ≤ x ≤ 2π - AQA - A-Level Maths Pure - Question 7 - 2022 - Paper 1

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:

1. Approaches infinity as it approaches the asymptote from the left.
2. 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.