Analyzing Cubic Function End-Behavior

00:00 4 marks

Consider the cubic function \(f(x) = ax^3 + bx^2 + cx + d\). Discuss how the sign of the coefficient \(a\) affects the end behavior of the function. Provide a brief explanation of why the tails of the graph are asymmetric.

Hint Hint

Hint

Recall that the sign of \(a\) dictates the direction in which the arms of the graph extend, and consider the impact of the odd degree of the polynomial.

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Answer Answer

Complete Answer

If \(a > 0\), as \(x \to \infty\), \(f(x) \to \infty\) and as \(x \to -\infty\), \(f(x) \to -\infty\). If \(a < 0\), then as \(x \to \infty\), \(f(x) \to -\infty\) and as \(x \to -\infty\), \(f(x) \to \infty\).

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