# 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: 57 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 &x+\frac{\pi}{2}&\le &2\pi \\&0-\frac{\pi}{2} &\le &x+\frac{\pi}{2}-\frac{\pi}{2} &\le &2\pi-\frac{\pi}{2} \\&-\frac{\pi}{2} &\le &x &\le &\frac{3\pi}{2} \end{array} Thus, one interval of the given function is $[-\frac{\pi}{2}, \frac{3\pi}{2}]$. Dividing this interval into four equal parts yield the key x-values $-\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi,$ and $\frac{3\pi}{2}$. To graph the given function, perform the following steps: (1) Create a table of values for the function $y=-3+2\sin{\left(x+\frac{\pi}{2}\right)}$ 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.) 