Graph y=7csc(x)

Problem

y=7*csc(x)

Solution

1. Identify the parent function and its properties. The function y=7*csc(x) is a transformation of y=csc(x) which is the reciprocal of y=sin(x)

2. Determine the vertical stretch. The coefficient 7 indicates a vertical stretch by a factor of 7 This means the local minima of the curves will be at y=7 and the local maxima will be at y=−7

3. Locate the vertical asymptotes. Since csc(x)=1/sin(x) the function is undefined where sin(x)=0 This occurs at x=n*π for any integer n

4. Find key points for one period. For the interval (0,2*π) the asymptotes are at x=0 x=π and x=2*π

• At x=π/2 sin(π/2)=1 so y=7*(1)=7

• At x=(3*π)/2 sin((3*π)/2)=−1 so y=7*(−1)=−7

5. Sketch the curves. Draw the vertical asymptotes. Between x=0 and x=π the graph is a U-shaped curve opening upward with a vertex at (π/2,7) Between x=π and x=2*π the graph is a U-shaped curve opening downward with a vertex at ((3*π)/2,−7)

Final Answer

y=7*csc(x)

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