Subjects algebra

Cubic End Behavior 983Bbb

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1. **State the problem:** Describe the end behavior of the cubic function whose graph passes through the origin and exhibits typical cubic end behavior. 2. **Recall the general behavior of cubic functions:** A cubic function generally has the form $$f(x) = ax^3 + bx^2 + cx + d$$ where $a \neq 0$. 3. **End behavior rules for cubic functions:** - If $a > 0$, as $x \to +\infty$, $f(x) \to +\infty$ and as $x \to -\infty$, $f(x) \to -\infty$. - If $a < 0$, as $x \to +\infty$, $f(x) \to -\infty$ and as $x \to -\infty$, $f(x) \to +\infty$. 4. **Apply to the given graph:** The graph passes through the origin and shows typical cubic end behavior with $f(x)$ increasing without bound as $x \to +\infty$ and decreasing without bound as $x \to -\infty$. 5. **Conclusion:** - As $x \to +\infty$, $f(x) \to +\infty$. - As $x \to -\infty$, $f(x) \to -\infty$. This matches the behavior of a cubic function with a positive leading coefficient. **Final answer:** As $x$ approaches infinity, $f(x)$ approaches $+\infty$. As $x$ approaches negative infinity, $f(x)$ approaches $-\infty$.