Problem 17 - College Algebra

Question

Determine the end behavior of the functions. $$f(x)=x^{4}$$

Step-by-Step Solution

Answer

The function \( f(x) = x^4 \) rises to infinity at both ends.

Step 1: Analyze the Leading Term

The function given is \( f(x) = x^4 \). The leading term of this polynomial function is \( x^4 \). To determine the end behavior, we need to consider this leading term because it dominates the behavior of the polynomial as \( x \) approaches positive or negative infinity.

Step 2: Determine the Leading Coefficient and Exponent

For the leading term \( x^4 \), the leading coefficient is \( 1 \) and the exponent is \( 4 \). Both of these values are positive, which is crucial when determining the end behavior. Specifically, the exponent \( 4 \) is an even number, affecting how the function behaves at the infinity limits.

Step 3: Evaluate End Behavior Based on the Exponent

Since the exponent is even, both ends of the function will either point upwards or downwards. Given that the leading coefficient is positive, \( x^4 \) will tend towards positive infinity as \( x \) approaches both positive infinity and negative infinity. This indicates that the graph of the function rises on both ends.

Step 4: State the End Behavior Conclusively

Thus, the end behavior of the function \( f(x) = x^4 \) is that as \( x \to +fty \), \( f(x) \to +fty \) and as \( x \to -fty \), \( f(x) \to +fty \).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole-number exponents. These functions can range from simple linear equations like \( f(x) = x + 1 \) to more complex ones involving multiple terms and higher powers such as \( f(x) = x^4 + 3x^3 - 2x^2 + x - 5 \). A key characteristic of polynomial functions is that they are continuous and smooth curves. This means, you can draw them without lifting your pencil off the paper.

Polynomial functions are important because they appear frequently in various fields, including physics, economics, and engineering. They are used to model a wide range of real-world phenomena. Analyzing these functions allows us to understand how their graphs behave, particularly at the extremes of the inputs, which is referred to as the end behavior of the function.

• Leading Term: The term in a polynomial with the highest degree (largest exponent).
• Degree: The highest power of the variable in the polynomial.
• Continuous Graph: No breaks, holes, or sharp edges in the curve.

Understanding polynomial functions lays the groundwork for analyzing their end behavior using concepts like leading term analysis.

Leading Term Analysis

Leading term analysis is a technique used to determine the overall shape and end behavior of a polynomial function. The leading term is the term with the highest exponent because it has the most significant impact on the function's behavior as \( x \) becomes very large or very small. For the function \( f(x) = x^4 \), the leading term is \( x^4 \), which is crucial to understanding the end behavior.

To perform leading term analysis, follow these steps:

• Identify the leading term in the polynomial.
• Determine the leading coefficient, which is the constant before the leading term.
• Note the exponent of the leading term and whether it is even or odd.

The leading coefficient and exponent help in sketching the function. A positive leading coefficient and an even exponent, like in \( x^4 \), means that as \( x \) approaches both positive and negative infinity, the resultant function value will approach positive infinity.

This analysis allows us to predict the general direction of the graph without actually plotting every point, making it a powerful tool for understanding polynomial functions.

Even Exponent Behavior

When analyzing polynomial functions, the behavior of even exponents is a key feature that determines the graph's directions at the ends. If we take a simple example of a function like \( f(x) = x^4 \), the even exponent \( 4 \) tells us something important about how the graph behaves as \( x \) becomes very large or very small.

An even exponent results in symmetrical end behavior. This means that both ends of the function's graph will point in the same direction. Here's how this works:

• If the leading coefficient is positive, like in \( x^4 \), both ends will rise to positive infinity.
• If the leading coefficient were negative, both ends would fall to negative infinity.

In contrast, odd exponents have different end behaviors where one direction will differ from the other. However, for an even exponent like in the function \( x^4 \), the symmetry holds, making even exponent behavior predictable and straightforward.

This predictable characteristic helps in quickly identifying the direction of the graph’s ends, easing the process of graphing and understanding polynomial functions.