Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 4 - Review Exercises - Page 190: 26

Answer

Refer to the graph below.
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Work Step by Step

RECALL: (1) The function $y=a\cdot \sec{x}$ has a period of $2\pi$. (2) Consecutive asymptotes of the secant function are $x=\frac{\pi}{2}$ and $x=\frac{3\pi}{2}$ One period of the given function is in the interval $[0, 2\pi]$. The guide function for secant is $y=\frac{1}{2}\cos{x}$. Divide this interval onto four equal parts to get the key x-values $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$. To graph the given function, perform the following steps: (1) Create a table of values for the guide function $y=\frac{1}{2}\cos{x}$ using the key x-values listed above. (Refer to the table below.) (2) Plot the points from the table of values and connect them using a dashed curve (as the curve will only serve as a guide). (3) Graph the consecutive vertical asymptotes listed above. (4) Sketch the graph of $y=\frac{1}{2}\sec{x}$ by drawing (i) a U-shaped curve below the x-axis and between the consecutive vertical asymptotes. (ii) a half U-shaped curve from the point $(0, 0.5)$ to the asymptote $x=\frac{\pi}{2}$. (iii) a half U-shaped curve from the asymptote $x=\frac{3\pi}{2}$ to the point $(2\pi, 0.5)$ . (Refer to the graph in the answer part above.)
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