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Chapter 4: Problem 40

Graph two periods of the given cosecant or secant function. $$y=-\frac{3}{2} \sec \pi x$$

Short Answer

Expert verified
To graph two periods of the function \(y=-\frac{3}{2} \sec \pi x\), one can graph the related function \(y=-\frac{3}{2} \cos \pi x\) first. Then, add vertical asymptotes where the cosine function equals zero, and sketch the secant function based on those asymptotes and the graph of the cosine function.

Step by step solution

01

Understand the basic secant function graph

A basic \(sec(x)\) function has undefined values (unreachable) at every \(x\) value where \(cos(x) = 0\). Therefore, it has vertical asymptotes at \(x = \frac{(2n+1)\pi}{2}\), where \(n\) is an integer. They usually have a period of \(2\pi\) and the range is \(y \leq -1\) or \(y \geq 1\).
02

Draw the related cosine function

The function \(y=-\frac{3}{2} \sec \pi x\) is equivalent to \(y=-\frac{3}{2} \cos \pi x\). The coefficient of \(x\), \(\pi\), indicates a horizontal compression of the graph by a factor of \(\frac{1}{\pi}\), or a change in the period from \(2\pi\) to \(2\). The negative sign gives a reflection of the graph across the x-axis. The factor of \(-\frac{3}{2}\) means a vertical stretch of the base graph by a factor of \(\frac{3}{2}\). Draw this function for 2 periods first.
03

Draw the secant graph based on the cosine graph

Now you can draw the graph of the given secant function based on the cosine graph. At points where the cosine function equals to zero, draw vertical asymptotes for the secant function. The secant graph will diverge at these points. For the points in between, the secant graph will leave from the cosine graph, reach a maximum or minimum, then approach the cosine graph again. Repeat this process over two periods of the function.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used a tangent function to model the average monthly temperature of New York City, where \(x=1\) represents January, \(x=2\) represents February, and so on.

Use a graphing utility to graph each pair of functions in the same viewing rectangle. Use a viewing rectangle that shows the graphs for at least two periods. $$y=4 \cos \left(2 x-\frac{\pi}{6}\right) \text { and } y=4 \sec \left(2 x-\frac{\pi}{6}\right)$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing one complete cycle of \(y=A \cos (B x-C)\) I find it easiest to begin my graph on the \(x\) -axis.

Use the keys on your calculator or graphing utility for converting an angle in degrees, minutes, and seconds \(\left(D^{\circ} M^{\prime} S^{\prime \prime}\right)\) into decimal form, and vice versa. Convert each angle to \(D^{\circ} M^{\prime} S^{\prime \prime}\) form. Round your answer to the nearest second. $$50.42^{\circ}$$

Use a vertical shift to graph one period of the function. $$y=-3 \sin 2 \pi x+2$$
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