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Chapter 4: Problem 48

Use a graphing utility to graph the function. (Include two full periods.) $$y=\frac{1}{3} \sec \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)$$

Short Answer

Expert verified
Plot the points, draw the curves and asymptotes considering the transformations. The graph covers the range from \(-2\pi\) to \(2\pi\) on the x-axis to represent two full periods.

Step by step solution

01

Understand the Secant Function

First, it's crucial to understand that the secant function is the reciprocal of the cosine function. The secant function values are undefined wherever the cosine function is equal to zero. In a typical cosine graph, with a period from \(0\) to \(2\pi\), we have zeros at \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\). So the secant function is also undefined at these points.
02

Determine the Transformation of the Function

Analyzing the given function \(y=\frac{1}{3} \sec \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)\), we see several alterations or transformations. The alteration \(\frac{\pi x}{2}\) is stretching the graph horizontally by a factor of 2, \(\frac{\pi}{2}\) represents a horizontal shift to the left by 1 unit and the outside factor of \(\frac{1}{3}\) is compressing the graph vertically by a factor of 3.
03

Plot the Points

Considering the transformations, we can choose x-values such that we get the secant function where it is defined and not at the points where cosine is zero. The points to be plotted can be chosen based on these transformations. After plotting the points, the graph can be drawn by sketching U-shaped curves that approach but never touch the vertical lines at each undefined point.
04

Graph Two Full Periods

The original secant function has a period of \(2\pi\), but the function \(y=\frac{1}{3} \sec \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)\) is horizontally stretched by a factor of 2, therefore it has a period of \(4\pi\). So, to graph two full periods, the graph should span the range from \(-2\pi\) to \(2\pi\) on the x-axis.

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Most popular questions from this chapter

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) As \(x \rightarrow 0^{+},\) the value of \(f(x) \rightarrow\) (b) As \(x \rightarrow 0^{-},\) the value of \(f(x) \rightarrow\) (c) As \(x \rightarrow \pi^{+},\) the value of \(f(x) \rightarrow\) (d) As \(x \rightarrow \pi^{-},\) the value of \(f(x) \rightarrow\) $$f(x)=\cot x$$

Finding Arc Length Find the length of the are on a circle of radius \(r\) intercepted by a central angle \(\boldsymbol{\theta}\). $$r=15 \text { inches, } \theta=120^{\circ}$$

Airplane Ascent During takeoff, an airplane's angle of ascent is \(18^{\circ}\) and its speed is 275 feet per second. (a) Find the plane's altitude after 1 minute. (b) How long will it take for the plane to climb to an altitude of \(10,000\) feet?

The normal monthly high temperatures \(H\) (in degrees Fahrenheit) in Erie, Pennsylvania, are approximated by $$H(t)=56.94-20.86 \cos \left(\frac{\pi t}{6}\right)-11.58 \sin \left(\frac{\pi t}{6}\right)$$ and the normal monthly low temperatures \(L\) are approximated by $$L(t)=41.80-17.13 \cos \left(\frac{\pi t}{6}\right)-13.39 \sin \left(\frac{\pi t}{6}\right)$$ where \(t\) is the time (in months), with \(t=1\) corresponding to January (see figure). (Source: National Climatic Data Center (GRAPH CANNOT COPY). (a) What is the period of each function? (b) During what part of the year is the difference between the normal high and normal low temperatures greatest? When is it smallest? (c) The sun is northernmost in the sky around June \(21,\) but the graph shows the warmest temperatures at a later date. Approximate the lag time of the temperatures relative to the position of the sun.

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) As \(x \rightarrow 0^{+},\) the value of \(f(x) \rightarrow\) (b) As \(x \rightarrow 0^{-},\) the value of \(f(x) \rightarrow\) (c) As \(x \rightarrow \pi^{+},\) the value of \(f(x) \rightarrow\) (d) As \(x \rightarrow \pi^{-},\) the value of \(f(x) \rightarrow\) $$f(x)=\csc x$$
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