Problem 199 - Precalculus

Question

For the following exercises, make a table to confirm the end behavior of the function. $$f(x)=x^{4}-5 x^{2}$$

Step-by-Step Solution

Answer

The end behavior is: as \(x\to\pm\infty\), \(f(x)\to+\infty\).

Step 1: Determine the Function's Degree and Leading Coefficient

Examine the given polynomial function: \[ f(x) = x^4 - 5x^2 \]The degree of the polynomial is the highest power of \(x\), which is 4 in this case. The leading coefficient is the coefficient of this term, which is 1.

Step 2: Determine the End Behavior Based on Degree and Coefficient

For polynomial functions, end behavior is influenced by the highest degree and its leading coefficient. Since the degree is 4 (an even number) and the leading coefficient is positive (1), the end behavior of the function is: - As \(x\) approaches \(+\infty\), \(f(x)\) approaches \(+\infty\).- As \(x\) approaches \(-\infty\), \(f(x)\) approaches \(+\infty\).This is typical of even-degree polynomials with positive leading coefficients.

Step 3: Create a Table of Values

To further confirm the end behavior, select a few values of \(x\) and compute \(f(x)\):| \(x\) | \(f(x) = x^4 - 5x^2\) ||------|-------------------|| -3 | 0 || -2 | -4 || 0 | 0 || 2 | -4 || 3 | 0 |Observe that for large values of \(x\), both positive and negative, \(f(x)\) becomes very large, consistent with the end behavior determined earlier.

Step 4: Confirm the End Behavior Using the Table

From the table and calculations, it is confirmed that as \(x\) moves towards \(+\infty\) or \(-\infty\), \(f(x)\) indeed tends towards \(+\infty\), consistent with our predictions based on the degree and leading coefficient.

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving terms where each term is a product of a constant coefficient and a variable raised to a non-negative integer power. They look like:

• The simplest form is a single monomial, such as \(x^n\), where \(n\) is a non-negative integer.
• Combinations of monomials create polynomials, like \(x^4 - 5x^2\) in this case.

Polynomials can have different degrees, based on the highest power of the variable involved. Understanding a polynomial's structure helps you predict its behavior, especially as the variable inside it takes on larger and smaller values.

Degree and Leading Coefficient

The degree of a polynomial is determined by the highest power of the variable present. In the polynomial \(f(x) = x^4 - 5x^2\), the highest power of \(x\) is 4, making it a degree 4 polynomial.

The leading coefficient is the coefficient of the term with the highest degree. Here, the leading term is \(x^4\), with a coefficient of 1. Understanding both the degree and leading coefficient is crucial as they influence the polynomial's graph shape and end behavior.

• The degree shows how many times the graph can turn.
• The leading coefficient affects the direction of the graph.

Not all coefficients are important for end behavior, only the ones that pair with the highest power.

Even Degree Polynomials

Polynomials with even degrees have distinct characteristics: their ends point in the same direction. This is true because even degree terms always equal positive numbers as they involve even powers. For example, the square and fourth powers return positive results, whether you input a positive or a negative value of \(x\).

In our polynomial \(x^4 - 5x^2\), the highest power is 4, an even number, which suggests that both ends of the graph approach infinity.

• Positive leading coefficient: both ends of the graph will go up.
• Negative leading coefficient: both ends will go down.

Seeing this visually on a graph helps solidify understanding, with both arms either rising or falling.

Table of Values

Creating a table of values helps to visualize the polynomial function and confirm predictions about its behavior. By picking a range of \(x\) values, you can calculate corresponding \(f(x)\) values.
Here's how you do it:

• Choose different values for \(x\), including both negative and positive.
• Substitute each \(x\) into the polynomial to find \(f(x)\).

For instance:

• If \(x = -3\), then \(f(x) = 0\).
• If \(x = 2\), then \(f(x) = -4\).

A table offers a clear view of key points and trends, confirming mathematical analysis about how a polynomial behaves at different intervals.

End Behavior Analysis

Analyzing end behavior is crucial to understanding how a polynomial behaves as inputs grow extremely large or small. It's largely dictated by the polynomial's degree and leading coefficient.

From the step-by-step exercise:

• As \(x \to +\infty\): For an even degree and positive leading coefficient, \(f(x)\) grows towards \(+\infty\).
• As \(x \to -\infty\): Similarly, \(f(x)\) again rises towards \(+\infty\).

This behavior arises because higher power terms dominate as \(x\) increases in magnitude. Recognizing this empowers you to predict a polynomial's graph shape without computing every point. When analyzing a polynomial, always consider these fundamental principles for accurate interpretations.