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: 30

Answer

Refer to the graph below.
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Work Step by Step

RECALL: The function $y=c+\csc{x}$ has: period = $\pi$ vertical shift= $|c|$, (upward when $c \gt0$, downward when $c\lt0$) consecutive vertical asymptotes: $x=0, \pi, $ and $2\pi$ The given function has $c=-1$. Thus, it has: period = $2\pi$ vertical shift = $|-1|=1$ One period of this function is in the interval $[0, 2\pi]$. Divide this interval into four equal parts to obtain the key x-values $\frac{\pi}{2}, \pi, \frac{3\pi}{2}$. Note that $\pi$ is a vertical asymptote of the function. To graph the given function, perform the following steps: (1) Create a table of values using the key x-values listed above. (Refer to the table below.) (2) Graph the vertical asymptotes listed above. (3) Draw a U-shaped curve between the asymptotes $x=0$ and $x=\pi$ whose vertex is at $(\frac{\pi}{2}, 0)$ . Draw an inverted-U-shaped curve between the asymptotes $x=\pi$ and $2\pi$ whose vertex is at $(\frac{3\pi}{2}, -2)$. (Refer to the graph in the answer part above.)
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