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Chapter 3: Problem 24

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=-11 x^{4}-6 x^{2}+x+3$$

Short Answer

Expert verified
The end behavior of the function \(f(x)=-11x^4-6x^2+x+3\) can be described as follows: as x approaches either positive or negative infinity, \(f(x)\) approaches negative infinity. This conclusion comes from the leading coefficient test which relates the end behavior of the function to the degree and leading coefficient of the function.

Step by step solution

01

Identify the Degree and Leading Coefficient

The degree of a polynomial is the highest power of x present in the polynomial. The leading coefficient of a polynomial is the coefficient of the term with the highest power. In this case, the degree is 4 and the leading coefficient is -11.
02

Apply the Leading Coefficient Test

The Leading Coefficient Test states that the end behavior of a polynomial function will depend on the degree and the sign of the leading coefficient. If the degree is even and the leading coefficient is positive, the end behavior will be as follows: as x approaches positive or negative infinity, f(x) approaches positive infinity. If the leading coefficient is negative, the end behavior will be opposite: as x approaches positive or negative infinity, f(x) approaches negative infinity. Since our function has an even degree and a negative leading coefficient, it means that as x approaches positive or negative infinity, f(x) will approach negative infinity.
03

Formulate the End Behavior

Putting the results from the Leading Coefficient Test into words, as x approaches positive infinity, \(f(x)\) approaches negative infinity. Also, as x approaches negative infinity, \(f(x)\) approaches negative infinity.

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