Graph y=-sec(x)

Problem

y=−sec(x)

Solution

1. Identify the parent function and its properties. The function y=sec(x) is the reciprocal of cos(x) meaning it has vertical asymptotes where cos(x)=0 which occurs at x=π/2+n*π for any integer n

2. Determine the effect of the negative sign, which represents a reflection across the xaxis. While the parent function y=sec(x) has a local minimum at (0,1) and a local maximum at (π,−1) the reflected function y=−sec(x) has a local maximum at (0,−1) and a local minimum at (π,1)

3. Locate the vertical asymptotes, which remain unchanged by the vertical reflection. These occur at x=−π/2 x=π/2 x=(3*π)/2 and so on.

4. Plot key points within one period. For the interval (−π/2,π/2) the graph opens downward from the maximum (0,−1) For the interval (π/2,(3*π)/2) the graph opens upward from the minimum (π,1)

5. Sketch the curves approaching the asymptotes. The range of the function is (−∞,−1]∪[1,∞)

Final Answer

y=−sec(x)

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