1. **Problem Statement:** We have two trigonometric functions given in the form $f(x) = a \sin[b(x - c)] + d$ and $y = a \cos[b(x - c)] + d$ with given maximum and minimum points. 2. **For the sine function:** Given points: maxima at $\left(-\frac{7\pi}{6}, 1.5\right)$ and $\left(\frac{5\pi}{6}, 1.5\right)$, minima at $\left(-\frac{\pi}{6}, -5.5\right)$ and $\left(\frac{11\pi}{6}, -5.5\right)$. 3. **Find amplitude $a$ and vertical shift $d$:** Amplitude $a = \frac{\text{max} - \text{min}}{2} = \frac{1.5 - (-5.5)}{2} = \frac{7}{2} = 3.5$. Vertical shift $d = \frac{\text{max} + \text{min}}{2} = \frac{1.5 + (-5.5)}{2} = \frac{-4}{2} = -2$. 4. **Period and $b$:** Distance between two consecutive maxima is period $T$. From $-\frac{7\pi}{6}$ to $\frac{5\pi}{6}$: $$T = \frac{5\pi}{6} - \left(-\frac{7\pi}{6}\right) = \frac{5\pi}{6} + \frac{7\pi}{6} = 2\pi$$ Period formula: $T = \frac{2\pi}{b} \Rightarrow b = 1$. 5. **Find phase shift $c$:** Maximum of $\sin$ function occurs at $x = c + \frac{\pi}{2b}$. Given maximum at $x = -\frac{7\pi}{6}$, so: $$-\frac{7\pi}{6} = c + \frac{\pi}{2} \Rightarrow c = -\frac{7\pi}{6} - \frac{\pi}{2} = -\frac{7\pi}{6} - \frac{3\pi}{6} = -\frac{10\pi}{6} = -\frac{5\pi}{3}$$ Since $c$ is phase shift, we want the minimum positive value of $c$ modulo $2\pi$: $$c_{positive} = 2\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3} \approx 1.05$$ 6. **Answer for sine function:** Minimum positive $c$ to nearest hundredth is $1.05$ (Option A). --- 7. **For the cosine function:** Given minimum at $\left(\frac{\pi}{2}, -2\right)$ and maximum at $\left(\pi, 8\right)$. 8. **Find amplitude $a$ and vertical shift $d$:** $$a = \frac{8 - (-2)}{2} = \frac{10}{2} = 5$$ $$d = \frac{8 + (-2)}{2} = \frac{6}{2} = 3$$ 9. **Find period and $b$:** Distance between minimum and maximum is $\frac{T}{2}$ for cosine. $$\frac{T}{2} = \pi - \frac{\pi}{2} = \frac{\pi}{2} \Rightarrow T = \pi$$ Period formula: $T = \frac{2\pi}{b} \Rightarrow b = \frac{2\pi}{T} = \frac{2\pi}{\pi} = 2$. 10. **Find phase shift $c$:** Cosine maximum occurs at $x = c$. Given maximum at $x = \pi$, so $c = \pi$. 11. **Equation of cosine function:** $$y = 5 \cos[2(x - \pi)] + 3$$ **Summary:** - Minimum positive $c$ for sine function is approximately $1.05$ (Option A). - Cosine function parameters: $a=5$, $b=2$, $c=\pi$, $d=3$.