Graph y=1/3*cot(5x)

Problem

y=1/3*cot(5*x)

Solution

1. Identify the parent function and its properties. The function is a transformation of y=cot(x) which 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 coefficient of x is k=5 The period P is calculated by dividing the natural period of the cotangent function by k

P=π/5

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

5*x=0⇒x=0

5*x=π⇒x=π/5

The asymptotes occur at x=(n*π)/5 for any integer n

4. Identify the x-intercepts, which occur halfway between the asymptotes. For the first interval (0,π/5) the intercept is at the midpoint.

x=(0+π/5)/2=π/10

5. Determine the vertical stretch or compression. The amplitude-like coefficient is a=1/3 This means the graph is vertically compressed. At the quarter-points of the period, the yvalues will be 1/3 and −1/3

x=π/20⇒y=1/3*cot(π/4)=1/3

x=(3*π)/20⇒y=1/3*cot((3*π)/4)=−1/3

6. Sketch the graph by drawing the vertical asymptotes at x=0 and x=π/5 plotting the intercept at (π/10,0) and drawing the decreasing cotangent curve through the points (π/20,1/3) and ((3*π)/20,−1/3)

Final Answer

y=1/3*cot(5*x)* has period π/5* and asymptotes at *x=(n*π)/5

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