1. **Problem Statement:** We need to write an equation for a polynomial function given its graph. 2. **Given Information:** The polynomial has x-intercepts at approximately $-3$, $-1$, $2$, and $4$. 3. **Step 1: Write the general form using roots** A polynomial with roots $r_1, r_2, r_3, r_4$ can be written as: $$y = a(x - r_1)(x - r_2)(x - r_3)(x - r_4)$$ Here, the roots are $-3, -1, 2, 4$, so: $$y = a(x + 3)(x + 1)(x - 2)(x - 4)$$ 4. **Step 2: Determine the leading coefficient $a$** Since the graph falls steeply after $x=4$ and the polynomial has degree 4 (even degree), the leading coefficient $a$ is likely positive. 5. **Step 3: Expand the polynomial (optional for clarity)** First, multiply pairs: $$(x + 3)(x + 1) = x^2 + 4x + 3$$ $$(x - 2)(x - 4) = x^2 - 6x + 8$$ 6. **Step 4: Multiply the two quadratics:** $$y = a(x^2 + 4x + 3)(x^2 - 6x + 8)$$ $$= a\left(x^2(x^2 - 6x + 8) + 4x(x^2 - 6x + 8) + 3(x^2 - 6x + 8)\right)$$ $$= a\left(x^4 - 6x^3 + 8x^2 + 4x^3 - 24x^2 + 32x + 3x^2 - 18x + 24\right)$$ $$= a\left(x^4 - 2x^3 - 13x^2 + 14x + 24\right)$$ 7. **Step 5: Choose $a=1$ for simplicity** The polynomial equation is: $$y = (x + 3)(x + 1)(x - 2)(x - 4) = x^4 - 2x^3 - 13x^2 + 14x + 24$$ **Final answer:** $$\boxed{y = (x + 3)(x + 1)(x - 2)(x - 4)}$$ This polynomial matches the given roots and general shape of the graph.