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Chapter 5: Problem 19

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result. \(y=\sec \pi x-3\)

Short Answer

Expert verified
To graph the function y = sec(蟺x) - 3, first remember the basic secant graph, then understand the effect of the 蟺 - it changes the period of the function to 2, and the '-3' shifts the whole graph 3 units down. Asymptotes are located at x = 1/2, 3/2, etc. and also at x = -1/2, -3/2, etc. Draw 'U' shapes or upside-down 'U' shapes - above 'U' shape for positive half-period and below 'U' shape for negative half-period - between asymptotes to complete the graph.

Step by step solution

01

Understand the Secant Function

By definition, the secant function is the reciprocal of cosine function. This means that wherever cosine has zeros, secant has vertical asymptotes, because dividing by zero is undefined. The cosine values range between -1 and 1, so for secant, the graph is above 1 or below -1.
02

Understand the effect of 蟺 and -3 in the function

If the functions were just sec(x) or cos(x), they would have a period of 2蟺. However, the presence of the coefficient 蟺 in our function sec(蟺x) changes the period to 2 (since 2蟺/蟺 = 2). Also, the graph of 'sec' or 'cos' usually passes through y=1 or y=-1, but the presence of '-3' shifts the whole graph 3 units downwards.
03

Plot the function

The function will have vertical asymptotes at x-values which make the cosine function zero, multiplied by the period, as it corresponds to the sec(蟺x) part. Therefore, the function will have vertical asymptotes at x = 1/2, 3/2, 5/2, etc. as well as at x = -1/2, -3/2, -5/2, etc. Between two asymptotes, the graph will look like a 'U' or an upside-down 'U', depending on whether we are in the positive period part or the negative one.

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Most popular questions from this chapter

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