If the leading term of a polynomial function is 3 x^3, which...

If the leading term of a polynomial function is $3 x^{3}$, which of the following could possibly be the graph of the polynomial? Explain. Consider what happens for large positive values of $x$ and for large negative values of $x$.

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Key Concepts

Leading Term

The leading term of a polynomial is the term with the highest power of the variable. It is crucial because it dominates the polynomial’s behavior for very large positive or negative values of x, making it key to understanding the overall shape of the graph.

End Behavior

End behavior describes how the graph of a polynomial function behaves as the input values approach positive or negative infinity. It determines the direction in which the graph heads on the far right and far left, which is primarily influenced by the leading term.

Degree of the Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. This concept is important because it affects the number of turning points of the function and how the graph behaves at the extremes of the x-axis. For instance, an odd degree implies that the ends of the graph head in opposite directions.

Leading Coefficient Test

The leading coefficient test uses the sign of the coefficient of the highest-degree term to predict the end behavior of a polynomial function. For an odd-degree polynomial with a positive leading coefficient, the function will approach positive infinity as x becomes very large and negative infinity as x becomes very small.

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