1. The problem asks to find the formula for the function $f$ given a hint that it can be expressed in the form $a(x - c)(x - d)$, which is a factored quadratic form. 2. The graph description indicates a quadratic curve opening downwards with vertex near $(5, 12)$, so the parabola has a maximum at $x=5$ and $f(5) = 12$. 3. The general form of a quadratic function in factored form is: $$f(x) = a(x - c)(x - d)$$ where $c$ and $d$ are the roots (x-intercepts) of the parabola. 4. Since the parabola opens downwards, $a < 0$. 5. To find $c$ and $d$, we need the x-intercepts. From the graph description, the parabola crosses the x-axis near $x=2$ and $x=8$ (approximate from vertex at 5 and symmetry). 6. So, $c=2$ and $d=8$. 7. Substitute into the formula: $$f(x) = a(x - 2)(x - 8)$$ 8. Use the vertex point $(5, 12)$ to find $a$: $$12 = a(5 - 2)(5 - 8) = a(3)(-3) = -9a$$ 9. Solve for $a$: $$a = -\frac{12}{9} = -\frac{4}{3}$$ 10. Therefore, the formula for $f$ is: $$f(x) = -\frac{4}{3}(x - 2)(x - 8)$$ This formula matches the graph's shape and vertex.