In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=11x^4−6x^2+x+3

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a method used to determine the end behavior of polynomial functions based on the degree and the leading coefficient of the polynomial. The degree indicates the highest power of x in the polynomial, while the leading coefficient is the coefficient of that term. This test helps predict whether the graph of the polynomial will rise or fall as x approaches positive or negative 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 shape and end behavior of the graph. For example, a polynomial of even degree will have the same end behavior on both sides, while an odd degree polynomial will have opposite end behaviors. In the given function, the degree is 4, indicating it is an even-degree polynomial.

End Behavior of Polynomials

End behavior refers to the behavior of the graph of a polynomial function as the input values (x) approach positive or negative infinity. This behavior is influenced by the degree and leading coefficient of the polynomial. For even-degree polynomials with a positive leading coefficient, the graph will rise to infinity on both ends, while a negative leading coefficient would cause it to fall on both ends. Understanding end behavior is essential for sketching the graph of the polynomial.