Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=11x⁴−6x²+x+3

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1. Identify the degree of the polynomial function. The degree is the highest power of \(x\) in the polynomial. For \(f(x) = 11x^{4} - 6x^{2} + x + 3\), the degree is 4.
2. Determine the leading coefficient, which is the coefficient of the term with the highest degree. Here, the leading coefficient is 11.
3. Recall the Leading Coefficient Test rules for end behavior: For even degree polynomials, if the leading coefficient is positive, both ends of the graph go up; if negative, both ends go down. For odd degree polynomials, if the leading coefficient is positive, the left end goes down and the right end goes up; if negative, the left end goes up and the right end goes down.
4. Since the degree is 4 (even) and the leading coefficient is 11 (positive), conclude that as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).
5. Summarize the end behavior: The graph rises to the right and rises to the left.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test uses the degree and leading coefficient of a polynomial to determine the end behavior of its graph. It states that the sign and parity (even or odd) of the leading term dictate how the function behaves as x approaches positive or negative infinity.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. It influences the shape and end behavior of the graph. For example, even-degree polynomials have the same end behavior on both sides, while odd-degree polynomials have opposite end behaviors.

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 by the leading term and helps predict whether the graph rises or falls on each end, which is essential for sketching or analyzing the function.