Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=10x^6-x^5+2x-2

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1. Identify the leading term of the polynomial function. The leading term is the term with the highest power of x, which is $10x^6$.
2. Determine the degree of the polynomial. The degree is the highest power of x in the polynomial, which is 6.
3. Since the degree is even and the leading coefficient (10) is positive, the end behavior of the polynomial is such that as $x \to \infty$, $f(x) \to \infty$ and as $x \to -\infty$, $f(x) \to \infty$.
4. Draw an end behavior diagram to visually represent this behavior. The graph will rise to infinity on both the left and right ends.
5. Summarize the end behavior: The graph of the polynomial function rises to infinity on both ends.

Key Concepts

End Behavior of Polynomials

The end behavior of a polynomial function describes how the function behaves as the input values (x) approach positive or negative infinity. This behavior is primarily determined by the leading term of the polynomial, which is the term with the highest degree. For example, in the polynomial f(x) = 10x^6 - x^5 + 2x - 2, the leading term is 10x^6, indicating that as x approaches infinity, f(x) will also approach infinity.

Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable in the polynomial expression. It plays a crucial role in determining the end behavior of the graph. In the given function f(x) = 10x^6 - x^5 + 2x - 2, the degree is 6, which is even. This means that both ends of the graph will either rise or fall together, depending on the sign of the leading coefficient.

Leading Coefficient Test

The leading coefficient test helps predict the end behavior of a polynomial function based on the sign of the leading coefficient and the degree of the polynomial. If the leading coefficient is positive and the degree is even, the graph rises on both ends. Conversely, if the leading coefficient is negative and the degree is even, the graph falls on both ends. In the case of f(x) = 10x^6, the positive leading coefficient indicates that the graph will rise as x approaches both positive and negative infinity.