Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Describe the right-hand and left-hand behavior of the graph of the polynomial function. $$f(x)=2 x^{5}-5 x+7.5$$

Step-by-Step Solution

Answer

The right-hand behavior of the graph of the function \(f(x)=2 x^{5}-5 x+7.5 \) is that as \(x\) becomes very large, \(f(x) \) also becomes very large, and the left-hand behavior is that as \(x\) becomes very small, \(f(x) \) also becomes very small.

Step 1: Identify the Degree and the Leading Coefficient

The degree of a polynomial is the highest power of x in the function, and the leading coefficient is the coefficient of that highest-power term. In this case, the degree is 5 (from the term \(2x^{5}\)), and the leading coefficient is 2 (from the same term).

Step 2: Apply the End Behavior Rules

End behavior of polynomial functions can be determined from the degree and the leading coefficient. If the degree is odd and the leading coefficient is positive, as it is here, then as \(x\) approaches infinity (\(x \rightarrow \infty\)), \(f(x)\) also approaches infinity (\(f(x) \rightarrow \infty\)), and as \(x\) approaches negative infinity (\(x \rightarrow -\infty\)), \(f(x)\) approaches negative infinity (\(f(x) \rightarrow -\infty\)). This is what 'right-hand and left-hand behavior' are referring to.

Step 3: Statement of the End Behavior

In light of the above discussions, for the given function \(f(x)=2 x^{5}-5 x+7.5\), as \(x\) approaches infinity (\(x \rightarrow \infty\)), \(f(x) \rightarrow \infty\) (right-hand behavior), and as \(x\) approaches negative infinity (\(x \rightarrow -\infty\)), \(f(x) \rightarrow -\infty\) (left-hand behavior). This simply means that the graph of the function rises to the right and falls to the left.

Key Concepts

End Behavior

The end behavior of a polynomial function refers to the direction in which the graph of the function moves as the input, or x, approaches incredibly large positive or negative values. Understanding this concept is essential for interpreting and predicting how a graph will appear on either end.

For any polynomial, the end behavior is decided by two main components: the degree of the polynomial and the leading coefficient. Depending on these factors, the graph might rise or fall on each side. Here are some general rules to remember:

• If the degree of the polynomial is odd and the leading coefficient is positive, as it is in our example, the function rises to the right ( f(x) ightarrow "+inf") and falls to the left ( f(x) ightarrow "-inf").
• If the degree is odd but the leading coefficient is negative, the behavior switches: it falls to the right and rises to the left.
• If the degree is even and the leading coefficient is positive, the graph will rise on both ends.
• If the degree is even and the leading coefficient is negative, the graph will fall on both ends.

In essence, these rules help us predict the shape and direction of the polynomial graph without having to plot numerous points.

Degree of Polynomial

The degree of a polynomial is a critical aspect that influences its behavior. It pertains to the highest power of the variable in the polynomial expression. In our example, the polynomial function is given by \(f(x) = 2x^5 - 5x + 7.5\). Here, the highest power of \(x\) is 5, therefore, the degree is 5.

This degree can dictate several properties of the polynomial:

• The number of roots or solutions a polynomial can have is directly linked to its degree. A polynomial of degree \(n\) can potentially have up to \(n\) real roots.
• Polynomials of higher degrees are associated with more complex wave-like graphs, due to the additional turning points these could have.
• The degree also plays a role in determining the overall "smoothness" of the graph and how it approaches its end behavior.

Understanding the degree is essential for recognizing how complex the graph will be and how it behaves at extreme values of \(x\).

Leading Coefficient

Polynomials are defined not just by their degrees but also by their leading coefficients. The leading coefficient is crucial because it impacts the graph's overall stretch and the direction of its end behavior. In the polynomial \(f(x) = 2x^5 - 5x + 7.5\), the leading coefficient is the coefficient of the term with the highest power of \(x\), which is 2.

The sign and value of the leading coefficient can tell us a lot:

• If the leading coefficient is positive, the graph's end behavior will mirror the rules described earlier for the degree.
• A negative leading coefficient will essentially "flip" the graph's direction for an odd degree, while causing it to fall on both ends for an even degree.
• The magnitude of the leading coefficient can also affect the steepness of the graph. A larger number will generally lead to a steeper rise or fall.

Understanding the leading coefficient alongside the degree provides a full picture of how the polynomial function behaves across its domain.