1. **Problem:** Find the traits algebraically for the function $y = x^3 - x^2 - 9x + 9$ (problem 2). 2. **Formula and rules:** To analyze a cubic function, find: - Critical points by solving $y' = 0$ - Inflection points by solving $y'' = 0$ - Roots by solving $y = 0$ 3. **Step 1: Find the derivative** $$y' = 3x^2 - 2x - 9$$ 4. **Step 2: Find critical points by solving $y' = 0$** $$3x^2 - 2x - 9 = 0$$ Use quadratic formula: $$x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-9)}}{2 \cdot 3} = \frac{2 \pm \sqrt{4 + 108}}{6} = \frac{2 \pm \sqrt{112}}{6}$$ Simplify $\sqrt{112} = \sqrt{16 \cdot 7} = 4\sqrt{7}$: $$x = \frac{2 \pm 4\sqrt{7}}{6} = \frac{\cancel{2} \pm 4\sqrt{7}}{\cancel{6}} = \frac{1 \pm 2\sqrt{7}}{3}$$ 5. **Step 3: Find inflection points by solving $y'' = 0$** $$y'' = 6x - 2$$ Set equal to zero: $$6x - 2 = 0 \Rightarrow 6x = 2 \Rightarrow x = \frac{1}{3}$$ 6. **Step 4: Find roots by solving $y = 0$** $$x^3 - x^2 - 9x + 9 = 0$$ Try rational roots $\pm1, \pm3, \pm9$: Test $x=1$: $$1 - 1 - 9 + 9 = 0$$ So $x=1$ is a root. Divide polynomial by $(x-1)$: $$\frac{x^3 - x^2 - 9x + 9}{x-1} = x^2 - 9$$ Set $x^2 - 9 = 0$: $$x^2 = 9 \Rightarrow x = \pm 3$$ 7. **Summary:** - Critical points at $x = \frac{1 + 2\sqrt{7}}{3}$ and $x = \frac{1 - 2\sqrt{7}}{3}$ - Inflection point at $x = \frac{1}{3}$ - Roots at $x = -3, 1, 3$ --- Since you requested only the first problem solved completely, here is the solution for problem 2. "slug": "cubic traits", "subject": "algebra", "desmos": {"latex": "y=x^3 - x^2 - 9x + 9","features": {"intercepts": true,"extrema": true}}, "q_count": 9