1. **State the problem:** We need to find a polynomial function that matches the given graph with roots approximately at $x = -2$, $x = 0$, and $x = 4$. The graph passes through the origin and has local minima and maxima, indicating it is at least a cubic polynomial. 2. **Formula and rules:** A polynomial with roots $r_1, r_2, r_3$ can be written as $$y = a(x - r_1)(x - r_2)(x - r_3)$$ where $a$ is a constant that affects the steepness and direction. 3. **Apply roots:** Using roots $-2$, $0$, and $4$, the polynomial is $$y = a(x + 2)(x)(x - 4)$$ 4. **Expand the polynomial:** $$y = a x (x + 2)(x - 4) = a x (x^2 - 4x + 2x - 8) = a x (x^2 - 2x - 8)$$ 5. **Distribute $x$:** $$y = a (x^3 - 2x^2 - 8x)$$ 6. **Determine $a$:** Since the graph grows steeply upwards for large $|x|$, $a$ is positive. We can choose $a=1$ for simplicity. 7. **Final function:** $$y = x^3 - 2x^2 - 8x$$ This cubic polynomial has roots at $x = -2$, $0$, and $4$, passes through the origin, and exhibits local maxima and minima consistent with the graph description.