Question

Describe the right-hand and left-hand behavior of the graph of the polynomial function. $$f(x)=\frac{1}{3} x^{3}+5 x$$

Short answer

The right-hand behavior of the function \(f(x) = \frac{1}{3}x^3 + 5x\) is that, as \(x\) approaches positive infinity, \(f(x)\) also approaches positive infinity. The left-hand behavior of the function is that, as \(x\) approaches negative infinity, \(f(x)\) approaches positive infinity.

Step-by-step solution

Identify the degree

Looking at the given polynomial function \(f(x) = \frac{1}{3}x^3 + 5x\), the degree can be identified as the power of the highest term. Here, our degree is 3, which is an odd number.

Determine the leading coefficient

The leading coefficient in a polynomial function is the coefficient of the term with the highest degree. Here, the leading coefficient is \(\frac{1}{3}\), which is a positive number.

Determine the End Behavior

Given that the degree is an odd number and the leading coefficient is positive, the graph of the function will fall to the left and rise to the right. This means as \(x\) approaches negative infinity, \(f(x)\) approaches positive infinity, and as \(x\) approaches positive infinity, \(f(x)\) approaches positive infinity.

Key concepts

Polynomial Function

A polynomial function is an expression constructed from variables (also known as indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example is \(f(x) = \frac{1}{3}x^3 + 5x\), which is the function from our exercise.

Polynomial functions are among the simplest and most commonly used mathematical functions in both pure and applied mathematics. Understanding the basic properties of polynomial functions, such as the degree, leading coefficient, and end behavior, is crucial for graph analysis and solving various applied problems.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. It is a simple yet informative attribute because it tells us quite a lot about the behavior of the function, especially for large values of \(x\). In \(f(x) = \frac{1}{3}x^3 + 5x\), the degree is 3 because the highest exponent of \(x\) is 3.

The degree can help predict the number of roots and the possible number of extremums (maximum and minimum points) the graph can have. For example, a third-degree polynomial will have at most three roots and up to two extremums.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest power of the variable. It is 'leading' because it is the coefficient of the first term when the polynomial is written in standard form, that is, terms are written in descending powers.

In the polynomial \(f(x) = \frac{1}{3}x^3 + 5x\), the leading coefficient is \(\frac{1}{3}\). This value heavily influences the graph's end behavior. If the leading coefficient is positive, and the degree is odd, the right side of the graph will rise, while a negative leading coefficient would mean the right side falls.

Graph Analysis

Analyzing the graph of a polynomial involves considering its degree, leading coefficient, and other factors to predict its shape and behavior. With graph analysis, we can infer the end behavior, the number of turning points, and whether it crosses or touches the x-axis at its roots.

In our exercise, analyzing the graph of \(f(x)\), we observe that as \(x\) becomes large, the \(x^3\) term dominates, and thus the behavior is determined by \(x^3\). Since our leading coefficient is positive and the degree is odd, the graph falls to the left and rises to the right, making it symmetrical with respect to the origin in terms of its end behavior.