directions: graph each function and identify its key characteristics.\n9. $f(x) = x^3 - 10x^2 + 27x…

Explanation:

Step1: Identify domain and range

For any odd-degree polynomial, the domain and range are all real numbers. $$D: (-\infty, \infty), R: (-\infty, \infty)$$

Step2: Find zeros by factoring

Factor $f(x) = x^3 - 10x^2 + 27x - 18$ using the Rational Root Theorem. $$f(x) = (x-1)(x-3)(x-6) \implies x = 1, 3, 6$$

Step3: Determine relative extrema

Find the derivative $f'(x) = 3x^2 - 20x + 27$ and solve $f'(x) = 0$. $$x = \frac{20 \pm \sqrt{400 - 324}}{6} = \frac{20 \pm \sqrt{76}}{6} \approx 1.87, 4.80$$

Step4: Calculate extrema values

Evaluate $f(1.87) \approx 2.11$ (Max) and $f(4.80) \approx -8.26$ (Min). $$\text{Rel. Max} \approx (1.87, 2.11), \text{Rel. Min} \approx (4.80, -8.26)$$

Step5: Determine end behavior

Since the leading coefficient is positive and the degree is odd: $$\text{As } x \to -\infty, f(x) \to -\infty; \text{As } x \to \infty, f(x) \to \infty$$

Step6: Identify intervals of increase/decrease

Based on the critical points found in Step 3: $$\text{Inc: } (-\infty, 1.87) \cup (4.80, \infty); \text{Dec: } (1.87, 4.80)$$

Answer:

Domain: $(-\infty, \infty)$ Range: $(-\infty, \infty)$ Rel. Maximum(s): $\approx (1.87, 2.11)$ Rel. Minimum(s): $\approx (4.80, -8.26)$ End Behavior: As $x \to -\infty, f(x) \to -\infty$; As $x \to \infty, f(x) \to \infty$ Inc. Intervals: $(-\infty, 1.87) \cup (4.80, \infty)$ Dec. Intervals: $(1.87, 4.80)$ Zeros: $1, 3, 6$