Graph y=3sec(x)

Problem

y=3*sec(x)

Solution

1. Identify the parent function and its properties. The function y=3*sec(x) is a transformation of y=sec(x) which is the reciprocal of y=cos(x)

2. Determine the vertical stretch. The coefficient 3 indicates a vertical stretch by a factor of 3 While sec(x) has a range of (−∞,−1]∪[1,∞) the range of y=3*sec(x) is (−∞,−3]∪[3,∞)

3. Locate the vertical asymptotes. Since sec(x)=1/cos(x) the function is undefined where cos(x)=0 These asymptotes occur at x=π/2+n*π for any integer n

4. Find the local extrema. The relative minima and maxima occur where cos(x) has its peaks and valleys. At x=0 y=3*sec(0)=3*(1)=3 At x=π y=3*sec(π)=3*(−1)=−3

5. Sketch the curves. Draw the vertical asymptotes at x=−π/2,π/2,(3*π)/2 etc. Plot the points (0,3) and (π,−3) then draw the U-shaped branches that approach the asymptotes.

Final Answer

y=3*sec(x)

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