Understanding End Behavior: The Leading Term Test for Polynomial Functions

leading term test

The leading term test, also known as the leading coefficient test or the degree test, is a method used to determine the end behavior of a polynomial function

The leading term test, also known as the leading coefficient test or the degree test, is a method used to determine the end behavior of a polynomial function.

In general, a polynomial function is an expression in the form:
f(x) = a_nx^n + a_(n-1)x^(n-1) + … + a_1x + a_0

In this expression, the term with the highest degree is called the leading term, and the corresponding coefficient (a_n) is known as the leading coefficient.

The leading term test states that the end behavior of a polynomial function is determined by the sign of its leading coefficient and the degree of the polynomial.

1. If the degree of the polynomial (n) is even:
– If the leading coefficient (a_n) is positive, then as x approaches positive or negative infinity, the function f(x) also approaches positive infinity.
– If the leading coefficient (a_n) is negative, then as x approaches positive or negative infinity, the function f(x) approaches negative infinity.

2. If the degree of the polynomial (n) is odd:
– If the leading coefficient (a_n) is positive, then as x approaches positive infinity, the function f(x) approaches positive infinity, and as x approaches negative infinity, the function f(x) approaches negative infinity.
– If the leading coefficient (a_n) is negative, then as x approaches positive infinity, the function f(x) approaches negative infinity, and as x approaches negative infinity, the function f(x) approaches positive infinity.

To illustrate this, let’s consider an example. Suppose we have the polynomial function:
f(x) = 3x^4 – 2x^3 + 5x^2 – 7x + 1

In this case, the leading term is 3x^4 with a leading coefficient of 3. Since the degree of the polynomial is even, and the leading coefficient is positive, the end behavior of the function f(x) is as follows:
– As x approaches positive or negative infinity, f(x) also approaches positive infinity.

Using the leading term test can help us understand the general behavior of a polynomial function without having to draw the entire graph.

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