Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=9x⁶-3x⁴+x²-2

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1. Identify the degree and leading coefficient of the polynomial function. Here, the function is \(f(x) = 9x^6 - 3x^4 + x^2 - 2\). The degree is 6 (the highest power of \(x\)), and the leading coefficient is 9.
2. Recall that the end behavior of a polynomial is determined by the degree and the leading coefficient. Since the degree is even (6) and the leading coefficient is positive (9), the ends of the graph will both point upwards.
3. Use the general rule for end behavior: For even degree and positive leading coefficient, as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).
4. Represent this behavior in an end behavior diagram, which typically shows arrows indicating the direction of the graph as \(x\) approaches positive and negative infinity. Both arrows should point upwards.
5. Summarize the end behavior: The graph rises to the right and rises to the left, reflecting the polynomial's even degree and positive leading coefficient.

Key Concepts

End Behavior of Polynomial Functions

End behavior describes how the values of a polynomial function behave as x approaches positive or negative infinity. It is determined mainly by the leading term, which dominates the function for large absolute values of x. Understanding end behavior helps predict the general shape of the graph at its extremes.

Leading Term and Degree of a Polynomial

The leading term is the term with the highest power of x in a polynomial, and its degree is the exponent of that term. The degree and the sign of the leading coefficient dictate the end behavior of the polynomial. For example, an even degree with a positive leading coefficient means the graph rises on both ends.

End Behavior Diagrams

End behavior diagrams use arrows or symbols to visually represent how the graph behaves as x approaches infinity or negative infinity. These diagrams simplify understanding by showing whether the graph rises or falls on each end, based on the polynomial’s degree and leading coefficient.