# 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: 42

Refer to the blue graph below.

#### Work Step by Step

RECALL: The graph of $y=a \cdot \sin{(x-d)}$ has an amplitude of $|a|$ units and involves a horizontal shift of the parent function $y=\sin{x}$. The shift is $d$ units to the right when $d \gt 0$ and $|d|$ units to the left when $d\lt0$. The given function has $a=3$ and $d=\frac{3\pi}{2}$, which is positive. Thus, the given function has an amplitude of $3$ and involves a $\frac{3\pi}{2}$-unit shift to the right of the function $y=a \cdot \sin{x}$. To graph the given function, perform the following steps: (1) Graph the function $y= 3 \sin{x}$ over a two-period interval, which is $[0, 4\pi]$. (Refer to the red graph in the answer part above.) (2) Shift the graph of the function in (1) above $\frac{3\pi}{2}$ units to the right. (Refer to the blue graph in the answer part above.)

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