Problem 25 - College Algebra

Question

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=-5 x^{4}+7 x^{2}-x+9$$

Step-by-Step Solution

Answer

Both ends of the graph point down, meaning as \(x\) gets infinitely large in either the positive or negative direction, \(f(x)\) will always tend towards positive infinity.

Step 1: Identify the Leading Coefficient and the Degree

Identify the leading term of the polynomial, which is \(-5x^4\). The leading coefficient of this term is \(-5\) and the degree of the polynomial is \(4\), which is even.

Step 2: Apply the Leading Coefficient Test

Since the leading coefficient is negative and the degree of the polynomial is even, the end behavior of the graph will be that both ends point downwards.

Step 3: Inference

Thus, as \(x\) approaches negative infinity, \(f(x)\) also approaches positive infinity. Similarly, as \(x\) approaches positive infinity, \(f(x)\) approaches positive infinity.

Key Concepts

Polynomial End Behavior

Understanding a polynomial's end behavior is essential for graphing the function and predicting how it behaves at extreme values of its variable, typically as the variable approaches positive or negative infinity. End behavior is influenced by the degree and the leading coefficient of the polynomial.

For example, consider the polynomial function given by
\(f(x) = -5x^{4} + 7x^{2} - x + 9\). To understand its end behavior, we refer to the leading coefficient test. The test tells us that, when dealing with even-degree polynomials like this one (with a degree of 4), the ends of the graph will either point in the same direction or in opposite directions based on the sign of the leading coefficient.

Since our leading coefficient is negative (\( -5\)), both ends of the graph of our polynomial will point downwards. This means that as \(x\) approaches positive and negative infinity, the function values head towards negative infinity. The information on end behavior is crucial for sketching the rough graph of the polynomial and predicting its long-term tendencies.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable that appears in the polynomial expression. It is essential as it helps predict the number of roots, and its value fundamentally influences the shape of the graph of the polynomial.

In our example, the polynomial \(f(x) = -5x^4 + 7x^2 - x + 9\) has the highest power of 4, making it a fourth-degree polynomial. An even-degree polynomial, such as this one, will have similar behavior at both ends of its graph; meaning, both arms of the graph will either rise or fall together. In the case of odd-degree polynomials, the ends will move in opposite directions.

Degree also correlates with the number of turning points the graph can have. For a 4th-degree polynomial, the graph can have at most 3 turning points. These characteristics determined by the degree are fundamental in graphing and analyzing polynomials.

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. In the polynomial \(f(x) = -5x^4 + 7x^2 - x + 9\), the leading term is \( -5x^4\), thus \( -5\) is the leading coefficient. The sign and value of the leading coefficient impact the graph significantly, especially its end behavior.

If the leading coefficient is positive and the degree is even, the ends of the graph will point upwards. Conversely, if it is negative, as in our example, the ends will point downwards. The coefficient's value, while affecting the steepness of the graph, doesn't change the overall direction of the ends. Additionally, when the degree is odd, a positive coefficient results in the end behavior where the right side of the graph points upwards, and the left side points downwards, and vice versa for a negative coefficient.