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

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 example, if the leading coefficient is positive and the degree is odd, the graph will rise to the right and fall to the left.

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. They can be represented in the form f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n is the leading coefficient and n is the degree of the polynomial. Understanding the structure of polynomial functions is essential for analyzing their behavior and characteristics.

End Behavior of Graphs

The end behavior of a graph refers to the direction in which the graph moves as the input values (x) approach positive or negative infinity. This behavior is crucial for sketching graphs and understanding the overall shape of polynomial functions. The end behavior is primarily determined by the leading coefficient and the degree of the polynomial, which dictate whether the graph will rise or fall at the extremes.