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

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Identify the degree of the polynomial. The degree is the highest power of the variable x, which in this case is 4.

Determine the leading coefficient. The leading coefficient is the coefficient of the term with the highest degree, which is -5 for the term \(-5x^4\).

Use the Leading Coefficient Test to determine the end behavior. Since the degree is even and the leading coefficient is negative, the end behavior of the graph is that both ends will point downwards.

Recall that for an even degree polynomial with a negative leading coefficient, as \(x \to \infty\), \(f(x) \to -\infty\) and as \(x \to -\infty\), \(f(x) \to -\infty\).

Summarize the end behavior: The graph of the polynomial function \(f(x) = -5x^4 + 7x^2 - x + 9\) will fall to the left and fall to the right.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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, even-degree polynomials have the same end behavior on both sides, while odd-degree polynomials have opposite end behaviors.

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 behaves far away from the origin, which is essential for sketching the graph and analyzing its characteristics.

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