Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=7+2x-5x^2-10x^4

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Identify the leading term of the polynomial. The leading term is the term with the highest power of x, which in this case is -10x^4.

Determine the degree of the polynomial. The degree is the highest power of x, which is 4.

Identify the leading coefficient, which is the coefficient of the leading term. Here, it is -10.

Use the degree and the leading coefficient to determine the end behavior. Since the degree is even and the leading coefficient is negative, the end behavior is that as x approaches positive or negative infinity, f(x) approaches negative infinity.

Draw an end behavior diagram to visually represent that as x goes to both positive and negative infinity, the graph of the polynomial will go downwards.

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Key Concepts

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

End Behavior of Polynomials

The end behavior of a polynomial function describes how the function behaves as the input values (x) approach positive or negative infinity. This behavior is primarily determined by the leading term of the polynomial, which is the term with the highest degree. Depending on the degree and the leading coefficient, the graph will either rise or fall at the ends.

Leading Coefficient Test

The leading coefficient test helps predict the end behavior of a polynomial function based on the sign and degree of the leading term. If the leading coefficient is positive and the degree is even, the ends of the graph will rise; if the degree is odd, one end will rise and the other will fall. Conversely, if the leading coefficient is negative, the behavior is reversed.

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 shape and end behavior of the graph. For example, a polynomial of even degree will have both ends of the graph going in the same direction, while a polynomial of odd degree will have ends going in opposite directions.

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