1. The problem is to find an equation for a sine graph with the following characteristics: midline $y = -2$, amplitude $1$, period $\frac{\pi}{2}$, and a maximum at $x = \frac{\pi}{8}$. 2. The general form of a sine function is $$f(x) = A \sin(B(x - C)) + D$$ where $A$ is the amplitude, $B$ affects the period, $C$ is the horizontal shift, and $D$ is the vertical shift (midline). 3. The amplitude $A$ is the distance from the midline to the maximum or minimum, so here $A = 1$. 4. The period $P$ is related to $B$ by the formula $$P = \frac{2\pi}{B}$$ Given $P = \frac{\pi}{2}$, solve for $B$: $$B = \frac{2\pi}{P} = \frac{2\pi}{\frac{\pi}{2}} = 4$$ 5. The midline $D$ is $-2$, so $D = -2$. 6. The maximum occurs at $x = \frac{\pi}{8}$. The sine function normally has a maximum at $\frac{\pi}{2}$, so we find the phase shift $C$ by solving: $$B(x - C) = \frac{\pi}{2}$$ Substitute $x = \frac{\pi}{8}$ and $B = 4$: $$4\left(\frac{\pi}{8} - C\right) = \frac{\pi}{2}$$ Simplify: $$\frac{\pi}{2} - 4C = \frac{\pi}{2}$$ Subtract $\frac{\pi}{2}$ from both sides: $$-4C = 0$$ Divide both sides by $-4$: $$\cancel{-4}C = \cancel{0} \Rightarrow C = 0$$ 7. Therefore, the phase shift $C = 0$. 8. Putting it all together, the equation is: $$f(x) = 1 \cdot \sin(4(x - 0)) - 2 = \sin(4x) - 2$$ 9. This matches the given equation and satisfies all the conditions of the problem.