Graph 2sec(x)

Problem

y=2*sec(x)

Solution

1. Identify the parent function and its properties. The function y=sec(x) is the reciprocal of y=cos(x) meaning it has vertical asymptotes where cos(x)=0

2. Determine the vertical asymptotes by finding where the cosine function is zero. These occur at x=π/2+n*π for any integer n

3. Identify the amplitude transformation of the related cosine graph. The coefficient 2 in y=2*sec(x) means the related cosine graph y=2*cos(x) has an amplitude of 2

4. Determine the range of the function. Since |sec(x)|≥1 the range of y=2*sec(x) is (−∞,−2]∪[2,∞)

5. Plot key points based on the peaks and valleys of y=2*cos(x) Local minima occur at (0,2) and (2*π,2) while a local maximum occurs at (π,−2)

6. Sketch the curves approaching the vertical asymptotes. The graph consists of U-shaped branches that open upward from the local minima and downward from the local maxima.

Final Answer

y=2*sec(x)

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