1. **State the problem:** Find the equation of a quadratic function in vertex form given the vertex at $(3,-6)$ and points $(1,0)$ and $(5,0)$ on the parabola. 2. **Recall the vertex form of a quadratic function:** $$y = a(x - h)^2 + k$$ where $(h,k)$ is the vertex. 3. **Substitute the vertex:** $$y = a(x - 3)^2 - 6$$ 4. **Use a point on the parabola to find $a$:** Using point $(1,0)$: $$0 = a(1 - 3)^2 - 6$$ $$0 = a( -2)^2 - 6$$ $$0 = 4a - 6$$ 5. **Solve for $a$:** $$4a = 6$$ $$a = \frac{6}{4}$$ $$a = \frac{3}{2}$$ 6. **Write the final equation:** $$y = \frac{3}{2}(x - 3)^2 - 6$$ 7. **Check with the other point $(5,0)$:** $$y = \frac{3}{2}(5 - 3)^2 - 6 = \frac{3}{2}(2)^2 - 6 = \frac{3}{2} \times 4 - 6 = 6 - 6 = 0$$ This confirms the equation is correct. **Final answer:** $$y = \frac{3}{2}(x - 3)^2 - 6$$