1. **State the problem:** We are given the x-intercepts of a parabola at $(-2,0)$ and $(3,0)$ and the vertex at $(0.5,-12.5)$. We need to find the equation of the parabola. 2. **Formula and rules:** The standard form of a parabola with roots $r_1$ and $r_2$ is: $$y = a(x - r_1)(x - r_2)$$ where $a$ is a constant that affects the parabola's width and direction. 3. **Substitute the roots:** Using the given x-intercepts $r_1 = -2$ and $r_2 = 3$, the equation becomes: $$y = a(x + 2)(x - 3)$$ 4. **Use the vertex to find $a$:** The vertex is at $(0.5, -12.5)$, so substitute $x=0.5$ and $y=-12.5$: $$-12.5 = a(0.5 + 2)(0.5 - 3)$$ Calculate inside the parentheses: $$-12.5 = a(2.5)(-2.5)$$ $$-12.5 = a(-6.25)$$ 5. **Solve for $a$:** $$a = \frac{-12.5}{-6.25} = 2$$ 6. **Write the final equation:** $$y = 2(x + 2)(x - 3)$$ 7. **Expand if desired:** $$y = 2(x^2 - x - 6) = 2x^2 - 2x - 12$$ **Final answer:** $$y = 2x^2 - 2x - 12$$