Cubic Intercepts 3Afb76

1. **State the problem:** We are given the cubic function $$P(x) = x^3 - 3x^2 - 18x$$ and need to find: - The y-intercept - The x-intercepts - The behavior of $y$ as $x \to \infty$ and $x \to -\infty$ 2. **Find the y-intercept:** The y-intercept occurs when $x=0$. $$P(0) = 0^3 - 3 \cdot 0^2 - 18 \cdot 0 = 0$$ So, the y-intercept is at the point $(0,0)$. 3. **Find the x-intercepts:** The x-intercepts occur where $P(x) = 0$. Set the function equal to zero: $$x^3 - 3x^2 - 18x = 0$$ Factor out the common factor $x$: $$x(x^2 - 3x - 18) = 0$$ This gives one root immediately: $$x = 0$$ Now solve the quadratic: $$x^2 - 3x - 18 = 0$$ Use the quadratic formula: $$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-18)}}{2 \cdot 1} = \frac{3 \pm \sqrt{9 + 72}}{2} = \frac{3 \pm \sqrt{81}}{2}$$ Since $\sqrt{81} = 9$, we have: $$x = \frac{3 \pm 9}{2}$$ Calculate both roots: - $$x = \frac{3 + 9}{2} = \frac{12}{2} = 6$$ - $$x = \frac{3 - 9}{2} = \frac{-6}{2} = -3$$ So the x-intercepts are at $x = 0$, $x = 6$, and $x = -3$. 4. **Behavior as $x \to \infty$:** For large positive $x$, the highest degree term $x^3$ dominates. Since the coefficient of $x^3$ is positive, as $x \to \infty$, $P(x) \to \infty$. 5. **Behavior as $x \to -\infty$:** For large negative $x$, again the $x^3$ term dominates. Since $x^3$ is negative for negative $x$ and the coefficient is positive, as $x \to -\infty$, $P(x) \to -\infty$. **Final answers:** - Y-intercept: $(0,0)$ - X-intercepts: $x = -3, 0, 6$ - As $x \to \infty$, $y \to \infty$ - As $x \to -\infty$, $y \to -\infty$