# 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: 52 The parent function of $y=1-\frac{2}{3}\sin{(\frac{3}{4}x)}$ is $y=\sin{x}$. The parent function $y=\sin{x}$ has a period of $2\pi$ so one period of its graph is in the interval $[0,2\pi]$ Note that the period of $y=d+\sin{(bx)}$ is $\frac{2\pi}{b}$. The given function has $b=\frac{3}{4}$ so its period is $\frac{2\pi}{\frac{3}{4}}=\frac{8\pi}{3}$. This means that one period of the given function is in the interval $[0, \frac{8\pi}{3}]$. Dividing this interval into four equal parts yield the key x-values $0, \frac{2\pi}{3}, \frac{4\pi}{3}, 2\pi,$ and $\frac{8\pi}{3}$ To graph the given function, perform the following steps: (1) Create a table of values for $y=1-\frac{2}{3}\sin{(\frac{3}{4}x)}$ using the key x-values listed above. (Refer to the table below.) (2) Plot each point from the table of values and connect them using a sinusoidal curve to complete one period in the interval $[0,\frac{8\pi}{3}]$. (3) Extend the graph one more period by repeating the cycle in the interval $[\frac{8\pi}{3},\frac{16\pi}{3}]$. Refer to the graph in the answer part above. 