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

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a method used to determine the end behavior of polynomial functions based on the sign and degree of the leading term. The leading term is the term with the highest power of x, and its coefficient influences whether the graph rises or falls as x approaches positive or negative infinity. For even-degree polynomials, the ends of the graph will either both rise or both fall, while for odd-degree polynomials, one end will rise and the other will fall.

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. In the given polynomial, the degree is 4, which is even, indicating that the graph will have the same behavior at both ends. Understanding the degree helps predict how the polynomial will behave as x approaches infinity or negative infinity.

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 the polynomial in the question, since the leading coefficient is negative and the degree is even, the graph will fall on both ends, indicating that as x approaches both positive and negative infinity, the function values will decrease.