1. The problem asks to find the equations of a sine graph that has been horizontally shifted, specifically the closest left and right shifts. 2. The general form of a horizontally shifted sine function is: $$y = \sin(x - c)$$ where $c$ is the horizontal shift. If $c > 0$, the graph shifts to the right by $c$ units; if $c < 0$, it shifts to the left by $|c|$ units. 3. From the graph's x-axis labels, the sine wave appears shifted from the standard sine wave which has zeros at multiples of $\pi$. 4. To find the closest left shift, identify the shift $c$ such that the graph matches $y = \sin(x - c)$ and $c$ is negative but closest to zero. 5. To find the closest right shift, identify the shift $c$ such that $c$ is positive and closest to zero. 6. Since the sine function is periodic with period $2\pi$, shifts differing by $2\pi$ represent the same graph. 7. Suppose the shift is $c = -\frac{\pi}{2} = -1.57$ (approx) for the closest left shift. 8. For the closest right shift, add $2\pi$ to the left shift: $c = -\frac{\pi}{2} + 2\pi = \frac{3\pi}{2} = 4.71$ (approx). 9. Therefore, the equations are: Closest left shift: $$y = \sin\left(x - (-1.57)\right) = \sin(x + 1.57)$$ Closest right shift: $$y = \sin(x - 4.71)$$ 10. Rounded to two decimal places, the shifts are $-1.57$ and $4.71$ respectively. Final answers: Closest left shift: $y = \sin(x + 1.57)$ Closest right shift: $y = \sin(x - 4.71)$