Problem 23 - 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

Based on the Leading Coefficient Test, both ends of the graph of the polynomial function \( f(x) = 5x^4 + 7x^2 - x + 9 \) point upwards in the y-direction (i.e., the end behavior is such that as \( x \) approaches positive or negative infinity, \( f(x) \) approaches positive infinity).

Step 1: Identify Degree and Leading Coefficient

First, identify the degree (the highest power in the polynomial) and the leading coefficient (the coefficient of the term with the highest power). Here, the degree of the polynomial \( f(x) = 5x^4 + 7x^2 - x + 9 \) is 4 (even) and the leading coefficient is 5 (positive).

Step 2: Apply Leading Coefficient Test

Next, apply the Leading Coefficient Test. This test determines the end behavior of a polynomial function based on the degree and leading coefficient. Since the degree is even and the leading coefficient is positive, as \( x \) approaches positive or negative infinity, \( f(x) \) also approaches positive infinity.

Step 3: State End Behavior

Finally, state the end behavior of the polynomial function, based on the findings from the Leading Coefficient Test. Both ends of the graph will point upwards in the y-direction, indicating that as \( x \) approaches positive or negative infinity, \( f(x) \) approaches positive infinity.

Key Concepts

Polynomial End Behavior

The end behavior of a polynomial function describes how the function behaves as the input value, or 'x', approaches infinity or negative infinity. Understanding this concept is crucial for predicting the long-term trends of a polynomial's graph without computing exact values for every point.

For the polynomial function given by f(x) = 5x^4 + 7x^2 - x + 9, we analyze its end behavior by looking at the highest power term, because it grows the fastest as 'x' becomes very large or very small. If the highest power is even, the ends of the graph will either both point up (if the leading coefficient is positive) or both point down (if it's negative). Conversely, if the highest power is odd, one end of the graph will point up, and the other will point down, with the exact direction depending on the sign of the leading coefficient.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable 'x' that appears in the function. It is a fundamental characteristic that influences many aspects of the polynomial's behavior, including the end behavior, the number of possible turning points, and the number of roots or zeros.

In f(x) = 5x^4 + 7x^2 - x + 9, the variable 'x' is raised to the fourth power in the term with the highest degree, making this a fourth-degree polynomial. An important property of fourth-degree polynomials is that they can have up to three turning points and up to four x-intercepts. Also, the degree being even has specific implications for the end behavior, as mentioned earlier.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient attached to the term with the highest power of 'x'. This number plays a pivotal role in shaping the graph. It can affect the steepness of the graph and, along with the degree, determines the polynomial's end behavior. A positive leading coefficient, as in our example where it is 5, indicates that the graph will point up at its ends if the degree is even, or up on the right end and down on the left if the degree is odd.

In conjunction with the Leading Coefficient Test, the leading coefficient reveals that as 'x' moves towards infinity or negative infinity, the y-values of f(x) will move towards positive infinity when the leading coefficient is positive and the degree is even.

Graph of Polynomial Function

Visualizing the graph of a polynomial is a powerful tool for understanding its behavior. The graph provides a visual representation of all the points where y is a function of x according to the polynomial equation. When graphing, factors such as intercepts, turning points, and end behavior help sketch an accurate shape of the polynomial.

Using the previous example of the polynomial f(x) = 5x^4 + 7x^2 - x + 9, we'd expect a smooth curve starting low, reaching one or more high points, and then rising as it heads off to both sides of the graph. By identifying the leading coefficient and the degree, we can confidently say that the ends of the curve will rise towards positive infinity, providing a general ‘U’ shape, as this is characteristic for polynomials with a positive leading coefficient and an even degree.