1. **State the problem:** We are given the function $f(x) = -\frac{1}{30}(x+3)(x-27)$ and want to understand its properties. 2. **Formula and rules:** This is a quadratic function in factored form. The zeros (roots) occur where each factor equals zero: $x+3=0$ or $x-27=0$. 3. **Find the roots:** $$x+3=0 \implies x=-3$$ $$x-27=0 \implies x=27$$ 4. **Expand the function:** $$f(x) = -\frac{1}{30}(x+3)(x-27) = -\frac{1}{30}(x^2 - 27x + 3x - 81) = -\frac{1}{30}(x^2 - 24x - 81)$$ 5. **Distribute the coefficient:** $$f(x) = -\frac{1}{30}x^2 + \frac{24}{30}x + \frac{81}{30} = -\frac{1}{30}x^2 + \frac{4}{5}x + \frac{27}{10}$$ 6. **Find the vertex:** The vertex of a quadratic $ax^2 + bx + c$ is at $x = -\frac{b}{2a}$. Here, $a = -\frac{1}{30}$ and $b = \frac{4}{5}$. Calculate: $$x = -\frac{\frac{4}{5}}{2 \times -\frac{1}{30}} = -\frac{\frac{4}{5}}{-\frac{2}{30}} = -\frac{\frac{4}{5}}{-\frac{1}{15}} = -\frac{4}{5} \times -15 = 12$$ 7. **Find the vertex's y-coordinate:** $$f(12) = -\frac{1}{30}(12+3)(12-27) = -\frac{1}{30}(15)(-15) = -\frac{1}{30} \times -225 = 7.5$$ 8. **Summary:** - Roots at $x = -3$ and $x = 27$ - Vertex at $(12, 7.5)$ - Parabola opens downward (since $a < 0$) **Final answer:** The function $f(x) = -\frac{1}{30}(x+3)(x-27)$ has roots at $x = -3$ and $x = 27$, and a vertex at $(12, 7.5)$ with the parabola opening downward.