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: 42

Answer

Refer to the graph below.
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Work Step by Step

RECALL: The graph of the function $y=c+a\cdot \cos{[b(x-d)]}$ 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 $[0, 2\pi]$ vertical shift = $|c|$ (upward when $|c|\gt0$, downward when $c\lt0$) Write the given function in the form $y=c+a\cdot\cos{[b(x-d)]}$ by factoring out $\pi$ within the cosine function: $y=0+(-\frac{1}{2})\cdot \cos{[\pi(x-1)]}$ The given function has $a=-\frac{1}{2}, b=\pi,c=0$ and $d=1$ Thus, the graph of this function has: period = $\frac{2\pi}{\pi}=2$ amplitude = $|-\frac{1}{2}| = \frac{1}{2}$ phase shift = $|1| = 1$ , to the right vertical shift = 0 (none) Since $a$ is negative, the graph also involves a reflection about the x-axis f the parent function. One period of this function will be in the interval $[0-1, 2-1]=[-1, 1]$. Divide this interval into four equal parts to get the key x-values $-1, -0.5, 0, 0.5, 1$. 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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