1. **Problem Statement:** We have the function $y = a + b \sin\left(x + \frac{\pi}{3}\right)$ and are given that the maximum value of $y$ is $\frac{\sqrt{3}}{2}$ and the minimum value is $-\frac{\sqrt{3}}{2}$. The $x$-intercept is at $x = \pi$. We need to find the value of $b$ from the given options. 2. **Formula and Important Rules:** The general form of a sine function is $y = a + b \sin(x + c)$ where: - $a$ is the vertical shift (midline of the sine wave), - $b$ is the amplitude (distance from midline to max or min), - $c$ is the horizontal phase shift. The maximum value of $y$ is $a + b$ and the minimum value is $a - b$. 3. **Using the given max and min values:** $$\max y = a + b = \frac{\sqrt{3}}{2}$$ $$\min y = a - b = -\frac{\sqrt{3}}{2}$$ 4. **Find $a$ and $b$ by solving the system:** Add the two equations: $$ (a + b) + (a - b) = \frac{\sqrt{3}}{2} + \left(-\frac{\sqrt{3}}{2}\right) $$ $$ 2a = 0 \implies a = 0 $$ Subtract the min from the max: $$ (a + b) - (a - b) = \frac{\sqrt{3}}{2} - \left(-\frac{\sqrt{3}}{2}\right) $$ $$ 2b = \sqrt{3} \implies b = \frac{\sqrt{3}}{2} $$ 5. **Check the $x$-intercept at $x=\pi$:** At $x=\pi$, $y=0$ (since it's an intercept): $$ 0 = a + b \sin\left(\pi + \frac{\pi}{3}\right) = 0 + \frac{\sqrt{3}}{2} \sin\left(\frac{4\pi}{3}\right) $$ $$ \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} $$ So, $$ y = \frac{\sqrt{3}}{2} \times \left(-\frac{\sqrt{3}}{2}\right) = -\frac{3}{4} \neq 0 $$ This suggests the vertical shift $a$ is zero but the intercept is not zero at $x=\pi$ unless the problem is focusing on amplitude $b$ only. 6. **Answer:** The amplitude $b$ is $\frac{\sqrt{3}}{2}$, which corresponds to option ۱) $\frac{\sqrt{3}}{2}$. **Final answer:** $b = \frac{\sqrt{3}}{2}$