Graph y=cot(3x)

Problem

y=cot(3*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 standard period of the cotangent function by the absolute value of the coefficient of x

P=π/3

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

3*x=0⇒x=0

3*x=π⇒x=π/3

The asymptotes occur at x=(n*π)/3 for any integer n

4. Locate the x-intercepts by finding the midpoint between the asymptotes, where the cotangent function equals zero.

x=(0+π/3)/2=π/6

5. Identify key points to determine the shape of the curve. Evaluate the function at the quarter-period marks.
   At x=π/12

y=cot(3⋅π/12)=cot(π/4)=1

At x=π/4

y=cot(3⋅π/4)=cot((3*π)/4)=−1

6. Sketch the graph by drawing vertical asymptotes at x=0 and x=π/3 plotting the x-intercept at (π/6,0) and plotting the points (π/12,1) and (π/4,−1) to show the decreasing shape of the cotangent curve.

Final Answer

y=cot(3*x)* has period π/3* and asymptotes at *x=(n*π)/3

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