Graph y=3cot((pix)/2)

Problem

y=3*cot((π*x)/2)

Solution

1. Identify the parent function and its properties. The function is a transformation of y=cot(x) which has vertical asymptotes where the argument is a multiple of π

2. Determine the period of the function. The period P of a cotangent function y=a*cot(b*x) is calculated using P=π/|b|

P=π/π/2

P=2

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

(π*x)/2=n*π

x=2*n

The asymptotes occur at x=0,±2,±4,…

4. Identify the x-intercepts which occur halfway between the asymptotes.

x=(0+2)/2=1

The intercepts occur at x=…,−1,1,3,…

5. Determine the vertical stretch. The coefficient a=3 stretches the graph vertically. Key points occur at one-quarter and three-quarters of the period.

At *x=0.5,y=3*cot(π/4)=3*(1)=3

At *x=1.5,y=3*cot((3*π)/4)=3*(−1)=−3

6. Sketch the graph by drawing the vertical asymptotes at x=0 and x=2 plotting the intercept at (1,0) and plotting the points (0.5,3) and (1.5,−3) to show the decreasing shape of the cotangent curve.

Final Answer

y=3*cot((π*x)/2)

The graph has a period of 2 vertical asymptotes at x=2*n and passes through points (0.5,3) (1,0) and (1.5,−3)

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