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Chapter 4: Problem 36

Sketch the graph of the function. (Include two full periods.) $$y=-\sec \pi x+1$$

Short Answer

Expert verified
The graph of \(y=-\sec \pi x+1\) is a flipped version of the graph of the secant function, shifted up by 1 unit, with vertical asymptotes at places where it crosses the x-axis. The function completes a full cycle in an interval of 2.

Step by step solution

01

Understand the function

The function has the form \(y=a \sec (bx) + c\). Here, \(a\) is -1, \(b\) is \(\pi\), and \(c\) is 1. The number \(a\) flips the graph if it's negative and stretches it vertically if it's absolute value is more than 1. The number \(b\) affects the period of the function, while \(c\) shifts the graph up and down.
02

Identify the period

For secant function, the period is given by \(2\pi / |b|\). In this case, \(b=\pi\), so the period becomes \(2\pi / |\pi|\) which simplifies to 2.
03

Draw the graph

Now, for drawing the graph, first, draw the cosine function \(y=-\cos(\pi x)\) with the period of 2. Then, wherever the cosine function crosses the x-axis (which is at \(x=0.5, 1.5, 2.5 ...\)), draw vertical asymptotes because secant function isn't defined there. Next, sketch the secant function as continuous curve approaching each asymptote. Finally, add 1 to each value of the secant function to get the final graph.

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