1. **State the problem:** Analyze the polynomial function $$f(x) = -5x^3 + 3x^2 - 4x + 18$$ 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 $f(0) = 18$. So the y-intercept is $(0, 18)$. - **x-intercepts:** Set $f(x) = 0$ and solve: $$-5x^3 + 3x^2 - 4x + 18 = 0$$ This cubic may not factor easily, so approximate or use numerical methods to find roots. 4. **End behavior:** For a cubic polynomial with leading term $-5x^3$: - As $x \to +\infty$, $-5x^3 \to -\infty$, so $f(x) \to -\infty$. - As $x \to -\infty$, $-5x^3 \to +\infty$, so $f(x) \to +\infty$. 5. **Transformations:** The function is a cubic reflected over the x-axis (due to the negative leading coefficient), vertically stretched by factor 5, and shifted by the other terms. Final summary: - Degree: 3 - y-intercept: $(0, 18)$ - x-intercepts: roots of $-5x^3 + 3x^2 - 4x + 18 = 0$ - End behavior: $f(x) \to -\infty$ as $x \to +\infty$, $f(x) \to +\infty$ as $x \to -\infty$ - Transformations: reflection over x-axis and vertical stretch by 5.