1. **State the problem:** Analyze the polynomial function $$y = -2x^3 + 5x^2 + 4x - 12$$ for its degree, intercepts, end behavior, and transformations. 2. **Degree:** The degree of a polynomial is the highest power of $x$. Here, the highest power is 3, so the degree is 3. 3. **Intercepts:** - **y-intercept:** Set $x=0$ to find $y$. $$y = -2(0)^3 + 5(0)^2 + 4(0) - 12 = -12$$ So, the y-intercept is $(0, -12)$. - **x-intercepts:** Set $y=0$ and solve for $x$: $$0 = -2x^3 + 5x^2 + 4x - 12$$ This cubic may be solved by factoring or numerical methods. 4. **End behavior:** For a cubic polynomial with leading coefficient $-2$ (negative), as $x \to \infty$, $y \to -\infty$, and as $x \to -\infty$, $y \to \infty$. 5. **Transformations:** The function is a cubic scaled by $-2$, shifted by the other terms $5x^2 + 4x - 12$. The negative leading coefficient reflects the graph over the x-axis compared to $y = x^3$. **Summary:** - Degree: 3 - y-intercept: $(0, -12)$ - x-intercepts: roots of $-2x^3 + 5x^2 + 4x - 12 = 0$ - End behavior: $y \to -\infty$ as $x \to \infty$, $y \to \infty$ as $x \to -\infty$ - Transformations: vertical stretch by 2, reflection over x-axis, plus quadratic and linear shifts.