In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $f\left(x\right)=5x^4+7x^2-x+9$

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, if the leading coefficient is positive, the ends of the graph will rise; if negative, the ends will fall. For odd-degree polynomials, the ends will behave oppositely.

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. The degree of the polynomial determines the maximum number of x-intercepts and the overall shape of the graph.

End Behavior of Graphs

End behavior refers to the behavior of the graph of a function as the input values (x) approach positive or negative infinity. Understanding end behavior is crucial for sketching graphs and analyzing functions, as it provides insight into how the function behaves outside the visible range. For polynomial functions, the end behavior is primarily influenced by the leading term, which dictates whether the graph rises or falls at the extremes.