Graph -csc(x)

Problem

y=−csc(x)

Solution

1. Identify the parent function and its properties. The function is y=csc(x) which is the reciprocal of sin(x) It has vertical asymptotes where sin(x)=0 which occur at x=n*π for any integer n

2. Determine the effect of the negative sign. The negative sign in front of the function, y=−csc(x) represents a reflection across the xaxis.

3. Locate the key points and asymptotes. For the parent function csc(x) on the interval (0,2*π) there are local minima at (π/2,1) and local maxima at ((3*π)/2,−1) After the reflection, these become a local maximum at (π/2,−1) and a local minimum at ((3*π)/2,1)

4. Sketch the curves between the asymptotes. The vertical asymptotes remain at x=0 x=π and x=2*π In the interval (0,π) the graph opens downward from the vertex (π/2,−1) In the interval (π,2*π) the graph opens upward from the vertex ((3*π)/2,1)

Final Answer

y=−csc(x)

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