Question: User: 494e Identify the expanded equation of a sketched cubic graph. The graph of $y = ax^3 + bx^2 + cx + d$ is sketched below. 144 -6 -3 4 Find the values of $a$, $b$, $c$ and $d$. a = b = c = d = Graph shape: cubic curve crossing the x-axis at $x = -6$, $x = -3$, and $x = 4$; it crosses the y-axis at $y = 144$ and has left end up, right end down, so position_hint = center.

1. **State the problem:** We need to find the coefficients $a$, $b$, $c$, and $d$ of the cubic polynomial $$y = ax^3 + bx^2 + cx + d$$ given the roots and a point on the graph. 2. **Use the roots to write the factored form:** Since the cubic crosses the x-axis at $x = -6$, $x = -3$, and $x = 4$, these are the roots. The polynomial can be expressed as $$y = a(x + 6)(x + 3)(x - 4)$$ where $a$ is a constant to be determined. 3. **Use the y-intercept to find $a$:** The graph crosses the y-axis at $y = 144$, which means when $x=0$, $y=144$. Substitute $x=0$: $$144 = a(0 + 6)(0 + 3)(0 - 4) = a \times 6 \times 3 \times (-4) = a \times (-72)$$ 4. **Solve for $a$:** $$a = \frac{144}{-72} = -2$$ 5. **Expand the factored form:** First expand $(x + 6)(x + 3)$: $$ (x + 6)(x + 3) = x^2 + 3x + 6x + 18 = x^2 + 9x + 18 $$ Now multiply by $(x - 4)$: $$ (x^2 + 9x + 18)(x - 4) = x^3 - 4x^2 + 9x^2 - 36x + 18x - 72 = x^3 + 5x^2 - 18x - 72 $$ 6. **Multiply by $a = -2$:** $$ y = -2(x^3 + 5x^2 - 18x - 72) = -2x^3 - 10x^2 + 36x + 144 $$ 7. **Identify coefficients:** $$ a = -2, \quad b = -10, \quad c = 36, \quad d = 144 $$ **Final answer:** $$ y = -2x^3 - 10x^2 + 36x + 144 $$