In Exercises 10–13, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] f(x) = x^6 -6x^4 + 9x^2

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. If the leading coefficient is positive and the degree is even, the ends of the graph will rise in both directions. Conversely, if the leading coefficient is negative and the degree is even, the ends will fall in both directions. For odd degrees, the ends will behave oppositely based on the sign of the leading coefficient.

Degree of a Polynomial

The degree of a polynomial is the highest power 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 degree 6, like f(x) = x^6 - 6x^4 + 9x^2, indicates that the graph will have certain symmetry and behavior at the extremes, specifically that it will rise to positive infinity as x approaches both positive and negative infinity.

End Behavior of Graphs

End behavior refers to the behavior of the graph of a function as the input values approach positive or negative infinity. Understanding end behavior is essential for sketching graphs and predicting how they will behave outside the visible range. For polynomials, the end behavior is primarily influenced by the degree and leading coefficient, which dictate whether the graph rises or falls at the ends.