Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=3+2x-4x²-5x¹⁰

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1. Identify the degree and leading coefficient of the polynomial function. The given function is \(f(x) = 3 + 2x - 4x^2 - 5x^{10}\). The term with the highest power of \(x\) is \(-5x^{10}\), so the degree is 10 and the leading coefficient is \(-5\).
2. Determine the end behavior based on the degree and leading coefficient. Since the degree is even (10) and the leading coefficient is negative (\(-5\)), the ends of the graph will both point downwards.
3. Express the end behavior in terms of limits: As \(x \to \infty\), \(f(x) \to -\infty\) and as \(x \to -\infty\), \(f(x) \to -\infty\).
4. Draw or visualize the end behavior diagram: both ends of the graph go down towards negative infinity.
5. Summarize the end behavior: The graph falls to negative infinity on both the left and right ends because the leading term dominates the behavior for very large positive and negative values of \(x\).

Key Concepts

Polynomial Functions and Degree

A polynomial function is an expression consisting of variables raised to whole-number exponents and their coefficients. The degree of the polynomial is the highest exponent of the variable, which largely determines the shape and end behavior of its graph.

Leading Term and Leading Coefficient

The leading term of a polynomial is the term with the highest degree, and its coefficient is the leading coefficient. These determine the end behavior of the polynomial's graph, indicating how the function behaves as x approaches positive or negative infinity.

End Behavior of Polynomial Graphs

End behavior describes how the values of a polynomial function behave as x approaches infinity or negative infinity. It depends on the degree and leading coefficient: even-degree polynomials with positive leading coefficients rise on both ends, while odd-degree polynomials have opposite end behaviors.