1. **State the problem:** We need to write an equation for a cosine function with a period of $3\pi$ and a phase shift of $+\frac{\pi}{3}$. 2. **Recall the general form of a cosine function:** $$y = A \cos(B(x - C)) + D$$ where $A$ is amplitude, $B$ affects the period, $C$ is the phase shift, and $D$ is the vertical shift. 3. **Period formula:** The period $T$ of a cosine function is related to $B$ by $$T = \frac{2\pi}{|B|}$$ Given $T = 3\pi$, solve for $B$: $$3\pi = \frac{2\pi}{|B|} \implies |B| = \frac{2\pi}{3\pi} = \frac{2}{3}$$ So, $B = \frac{2}{3}$ (taking positive for standard orientation). 4. **Phase shift:** Phase shift is given by $C$ in the formula $y = A \cos(B(x - C))$. Given phase shift is $+\frac{\pi}{3}$, so $$C = -\frac{\pi}{3}$$ because the formula uses $(x - C)$, so to shift right by $+\frac{\pi}{3}$, $C$ must be negative. 5. **Write the final equation:** Assuming amplitude $A=1$ and vertical shift $D=0$, the equation is $$y = \cos\left(\frac{2}{3}\left(x + \frac{\pi}{3}\right)\right)$$ This function has period $3\pi$ and phase shift $+\frac{\pi}{3}$.