Graph y=cot(4x)

Problem

y=cot(4*x)

Solution

1. Identify the parent function and its properties. The function is y=cot(b*x) where b=4 The parent function y=cot(x) has vertical asymptotes where sin(x)=0

2. Determine the period of the function. The period P of the cotangent function is calculated using the formula P=π/|b|

P=π/4

3. Find the vertical asymptotes by setting the argument of the cotangent function equal to n*π where n is an integer.

4*x=n*π

x=(n*π)/4

For one cycle starting at n=0 the asymptotes are at x=0 and x=π/4

4. Locate the x-intercepts by finding where the numerator of the cotangent (cosine) is zero. This occurs halfway between the asymptotes.

x=π/8

5. Identify key points to determine the shape. For y=cot(4*x) evaluate the function at the quarter-period marks.

x=π/16⇒y=cot(π/4)=1

x=(3*π)/16⇒y=cot((3*π)/4)=−1

6. Sketch the graph by drawing the vertical asymptotes at x=0,π/4,π/2,… plotting the x-intercepts at x=π/8,(3*π)/8,… and drawing the decreasing curves between the asymptotes.

Final Answer

y=cot(4*x)* has period π/4* and asymptotes at *x=(n*π)/4

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