1. **State the problem:** We need to find the quadratic function $f(x)$ in the form $a(x - c)(x - d)$ given the roots and vertex. 2. **Given information:** - Roots: $x=2$ and $x=8$ - Vertex: $(5,4)$ - The parabola opens downward, so $a<0$ 3. **Write the general form using roots:** $$f(x) = a(x - 2)(x - 8)$$ 4. **Use the vertex to find $a$:** The vertex $x$-coordinate is the midpoint of the roots: $$\frac{2 + 8}{2} = 5$$ Substitute $x=5$ and $f(5)=4$ into the function: $$4 = a(5 - 2)(5 - 8) = a(3)(-3) = -9a$$ 5. **Solve for $a$:** $$4 = -9a$$ $$\Rightarrow a = \frac{4}{-9} = -\frac{4}{9}$$ 6. **Write the final function:** $$f(x) = -\frac{4}{9}(x - 2)(x - 8)$$ This function has roots at 2 and 8, opens downward, and has vertex at (5,4).