Trigonometry (11th Edition) Clone

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

Chapter 4 - Graphs of the Circular Functions - Section 4.2 Translations of the Graphs of the Sine and Cosine Functions - 4.2 Exercises - Page 162: 59

Answer

Refer to the graph below.
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Work Step by Step

The given equation is already in the form $y=c+a\cdot \sin{[b(x-d)]}$. Find one interval of the given function whose length is one period: \begin{array}{ccccc} &0 &\le &2\left(x+\frac{\pi}{4}\right)&\le &2\pi \\&\frac{0}{2}&\le &\frac{2\left(x+\frac{\pi}{4}\right)}{2} &\le &\frac{2\pi}{2} \\&0 &\le &x+\frac{\pi}{4} &\le &\pi \\&0-\frac{\pi}{4} &\le &x+\frac{\pi}{4}-\frac{\pi}{4} &\le &\pi-\frac{\pi}{4} \\&-\frac{\pi}{4} &\le &x &\le &\frac{3\pi}{4} \end{array} Thus, one interval of the given function is $[-\frac{\pi}{4}, \frac{3\pi}{4}]$. Dividing this interval into four equal parts yield the key x-values $-\frac{\pi}{4}, 0, \frac{\pi}{4}, \frac{\pi}{2}, $ and $\frac{3\pi}{4}$. To graph the given function, perform the following steps: (1) Create a table of values for the function $y=\frac{1}{2}+\sin{\left(2(x+\frac{\pi}{4})\right)}$ using the key x-values listed above. (Refer to the table below.) (2) Plot each point in the table then connect them using a sinusoidal curve. (Refer to the attached graph in the answer part above.)
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