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