# How to draw the graph of y = cot(x - pi/4)

- Step 1: draw the graph of y = cot x
- when the value of x toward to zero from a number more than zero, the value of y is toward to positive infinity.
- when x = pi/4, y = cot(x) = cot(pi/4) = [cos(pi/4)]/[sin(pi/4)] = 1
- when x = pi/2, y = cot(x) = cot(pi/2) = [cos(pi/2)]/[sin(pi/2)] = 0/1 = 0
- when x = 3pi/4, y = cot(x) = cot(3pi/4) = [cos(3pi/4)]/[sin(3pi/4)] = [cos(pi - pi/4)]/[sin(pi - pi/4)] = [-cos(pi/4)]/[sin(pi/4)] = -1
- when the value of x toward to pi (pi = 3.14) from the number less than pi, the value of y is toward to negative infinity.

- Step 2: shift the graph of y= cot(x) right pi/4 units to get the graph of y = cot(x - pi/4)
- when the value of x toward to pi/4 = 0.785 (pi = 3.14) from a number large than pi/4, the value of y is toward to positive infinity.
- when x = pi/2 = 1.57 (pi = 3.14), y = cot(x - pi/4) = cot(pi/2 - pi/4) = cot(2pi/4 - pi/4) = cot(pi/4) = 1
- when x = 3pi/4, y = cot(x - pi/4) = cot(3pi/4 - pi/4) = cot(2pi/4) = cot(pi/2) = [cos(pi/2)]/[sin(pi/2)] = 0/1 = 0
- when x = pi, y = cot(x - pi/4) = cot(pi - pi/4) = [cos(pi - pi/4)]/[sin(pi - pi/4)] = [-cos(pi/4)]/[sin(pi/4)] = -1
- when the value of x toward to 5pi/4 = 3.925 (pi = 3.14) from a number less than 5pi/4, the value of y is toward to negative infinity.
- Note:
- In graph of y = cot(x), when x = pi/2, y = cot(x) = 0
- In graph of y = cot(x - pi/4), when x = 3pi/4, y = cot(x - pi/4) = 0
- 3pi/4 - pi/2 = 3pi/4 - 2pi/4 = pi/4
- Therefore, by shift the graph of y = cot(x) right pi/4 units to get the graph of y = cot(x - pi/4)