In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=5x^3+7x^2−x+9

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1. Identify the degree of the polynomial. The degree is the highest power of x, which is 3 in this case.
2. Determine the leading coefficient, which is the coefficient of the term with the highest degree. Here, the leading coefficient is 5.
3. Since the degree is odd (3) and the leading coefficient is positive (5), the end behavior of the polynomial can be determined.
4. For an odd degree polynomial with a positive leading coefficient, as x approaches positive infinity, f(x) also approaches positive infinity.
5. Similarly, as x approaches negative infinity, f(x) approaches negative infinity.

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.

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 overall shape and behavior of the graph. For example, a polynomial of odd degree will have different end behaviors compared to one of even degree, affecting how the graph behaves at the extremes.

End Behavior of a Polynomial

End behavior refers to the direction in which the graph of a polynomial function moves as the input values (x) approach positive or negative infinity. Understanding end behavior helps predict how the graph will behave outside the visible range, which is essential for sketching the graph accurately and analyzing its overall characteristics.