/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 Sketch one full period of the gr... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 26

Sketch one full period of the graph of each function. $$y=\frac{3}{4} \sec x$$

Short Answer

Expert verified
The graph of \(y=\frac{3}{4} \sec(x)\) is a vertically compressed version of the standard secant function graph. It spans a period of \(2\pi\) radians, starting and ending at \(y= \frac{3}{4}\) for \(x=0\) and \(x=2\pi\), respectively and has vertical asymptotes at \(x=\frac{\pi}{2}\) and \(x=\frac{3\pi}{2}\). The curve rises from \(y= \frac{3}{4}\) up to the vertical asymptote at \(x=\frac{\pi}{2}\), skips to negative infinity at the same x-value, rises back up to \(y= -\frac{3}{4}\) at \(x=\pi\), goes down again to the asymptote at \(x=\frac{3\pi}{2}\), and finally skips back up to \(y= \frac{3}{4}\) at \(x=2\pi\).

Step by step solution

01

Understand the base function

Secant (sec) is the reciprocal of the cosine (cos) function, i.e., \(sec(x) = \frac{1}{cos(x)}\), for all \(x\) where \(cos(x)\neq0\). We know that the cosine function repeats itself every \(2\pi\) radians and there are critical points at \(x=\frac{\pi}{2}\) and \(x=\frac{3\pi}{2}\), where the cosine function equals 0 and secant is undefined.
02

Apply the coefficient

The function in this exercise is \(y=\frac{3}{4} \sec x\). So, the specific function is a vertically compressed version of the standard secant curve. Every \(y\) value of the secant function is multiplied by \(\frac{3}{4}\) here. Hence, the highest and lowest points of our curve will be \(\frac{3}{4}\) times the usual secant function's value at these points.
03

Plot the function

At \(x=0\), \(cos(x)=1\) and hence \(sec(x) = \frac{1}{cos(x)} = 1\). Multiply this by the coefficient \(\frac{3}{4}\) to get \(y= \frac{3}{4}\). Similarly calculate the y values for critical points \(x=\frac{\pi}{2}\), \(x=\pi\), \(x=\frac{3\pi}{2}\) and \(x=2\pi\). You will get y values as undefined for \(x=\frac{\pi}{2}\) and \(x=\frac{3\pi}{2}\). Connect the points to sketch the graph of \(y=\frac{3}{4} \sec(x)\), remembering that at \(x=\frac{\pi}{2}\) and \(x=\frac{3\pi}{2}\), secant function has vertical asymptotes.

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Most popular questions from this chapter

When two sound waves have approximately the same frequency, the sound waves interfere with one another and produce phenomena called beats, which are heard as variations in the loudness of the sound. A piano tuner can use these phenomena to tune a piano. By striking a tuning fork and then tapping the corresponding key on a piano, the piano tuner listens for beats and adjusts the tension in the string until the beats disappear. Use a graphing utility to graph the functions in Exercises 83 to 86 which are based on beats. $$y=\sin (5 \pi x) \cdot \sin \left(-\frac{\pi}{2} x\right)$$

When two sound waves have approximately the same frequency, the sound waves interfere with one another and produce phenomena called beats, which are heard as variations in the loudness of the sound. A piano tuner can use these phenomena to tune a piano. By striking a tuning fork and then tapping the corresponding key on a piano, the piano tuner listens for beats and adjusts the tension in the string until the beats disappear. Use a graphing utility to graph the functions in Exercises 83 to 86 which are based on beats. $$y=\sin (13 \pi x) \cdot \sin \left(-\frac{\pi}{2} x\right)$$

Graph at least one full period of the function defined by each equation. $$y=\cos \frac{\pi x}{3}$$

Use a graphing utility to graph each function. $$y=x \cos x$$

Explain how to use the graph of \(y=f(x)\) to produce the graph of \(y=f(2 x) \).
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