Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=10x⁶-x⁵+2x-2

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1. Identify the leading term of the polynomial function. For the given function \(f(x) = 10x^6 - x^5 + 2x - 2\), the leading term is \$10x^6\( because it has the highest power of \)x$.
2. Determine the degree of the polynomial, which is the exponent of the leading term. Here, the degree is 6, an even number.
3. Look at the leading coefficient, which is the coefficient of the leading term. In this case, it is 10, a positive number.
4. Use the degree and leading coefficient to describe the end behavior: For an even degree and positive leading coefficient, as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).
5. Summarize the end behavior using an end behavior diagram or notation: both ends of the graph rise upwards, which can be represented as \(\uparrow \quad \quad \uparrow\).

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 very large or very small x-values.

Leading Term and Degree of a Polynomial

The leading term is the term with the highest power of x in a polynomial. The degree (the exponent of this term) and the leading coefficient (its coefficient) dictate the shape and direction of the graph's ends.

End Behavior Diagrams

End behavior diagrams use arrows or symbols to visually represent the direction of the graph as x approaches ±∞. They help summarize whether the graph rises or falls on each end based on the polynomial's degree and leading coefficient.