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: 38

Answer

Refer to the graph below.
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Work Step by Step

RECALL: The graph of the function $y=a\cdot \sin{[b(x-d)]}$ is a sinusoidal curve that has: period = $\frac{2\pi}{b}$ amplitude = $|a|$ phase shift = $|d|$ (to the left when $d\lt0$, to the right when $d\gt0$) one period is in the interval $[0, 2\pi]$ Write the given equation in the form $y= a \cdot \sin{[b(x-d)]}$ by factoring out $3$ within the sine function: $y=\sin{[3(x+\frac{\pi}{6})]}$ This function has $a=1, b=3,$ and $d=-\frac{\pi}{6}$ Thus, the graph of this function has: period = $\frac{2\pi}{3}$ amplitude = $|1| = 1$ phase shift = $|-\frac{\pi}{6}|=\frac{\pi}{6}$, to the left One period of this function will be in the interval $[0-\frac{\pi}{6}, \frac{2\pi}{3}-\frac{\pi}{6}]=[-\frac{\pi}{6}, \frac{\pi}{2}]$. Divide this interval into four equal parts to get the key x-values $-\frac{\pi}{6}, 0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}$. 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) Plot the points from the table of values and connect them using a sinusoidal curve. (Refer to the graph in the answer part above.)
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