Graph y=cot(2x)

Problem

y=cot(2*x)

Solution

1. Identify the parent function and its properties. The function is y=cot(B*x) where the parent function is y=cot(x) The parent function has a period of π and vertical asymptotes at x=n*π for any integer n

2. Determine the period of the transformed function. The period P is calculated by dividing the natural period of cotangent by the coefficient of x

P=π/2

3. Locate the vertical asymptotes by setting the argument of the cotangent function equal to the locations of the parent function's asymptotes (0 and π.

2*x=0⇒x=0

2*x=π⇒x=π/2

The asymptotes occur every π/2 units.

4. Find the x-intercepts by identifying where the cotangent function equals zero. This occurs halfway between the asymptotes.

x=(0+π/2)/2=π/4

5. Identify key points to determine the shape of the curve. For y=cot(2*x) evaluate the function at the quarter-period marks between x=0 and x=π/2
   At x=π/8

y=cot(2⋅π/8)=cot(π/4)=1

At x=(3*π)/8

y=cot(2⋅(3*π)/8)=cot((3*π)/4)=−1

6. Sketch the graph by drawing vertical dashed lines at x=0,π/2,π,… and plotting the points (π/8,1) (π/4,0) and ((3*π)/8,−1) then connecting them with a smooth curve that approaches the asymptotes.

Final Answer

y=cot(2*x)* has period π/2, asymptotes at *x=(n*π)/2, and x-intercepts at *x=π/4+(n*π)/2

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