Problem 23 - A Graphical Approach to College Algebra

Question

Graph each function in the standard viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then fill in the blank of the following sentence with either increasing or decreasing. Over the interval specified, this function is __________. $$f(x)=x^{5} ;(-\infty, \infty)$$

Step-by-Step Solution

Answer

Over the interval \((-\infty, \infty)\), this function is increasing.

Step 1: Understand the Function

The given function is a polynomial function of degree 5, specifically \( f(x) = x^5 \). Polynomial functions of this form have either an increasing or decreasing behavior based on the sign of the leading coefficient.

Step 2: Analyze the Leading Term

The leading term of the function \( f(x) = x^5 \) is \( x^5 \). Since the coefficient of \( x^5 \) is positive (1 in this case), the function will tend to rise as \( x \) increases.

Step 3: Determine Over the Entire Interval

For \( x^5 \), as we approach negative infinity (i.e., \( x \to -fty\)), the function will decrease as \( x \) becomes smaller, and as \( x \to +fty \), the function will keep increasing. Thus, considering the entire interval \( (-fty, fty)\), the function shows a general trend of increase from left to right.

Step 4: Conclude Based on Behavior

Since the behavior of the function \( f(x) = x^5 \) over the interval \( (-fty, fty) \) is rising from left to right, we conclude that the function is increasing throughout the specified interval.

Key Concepts

Understanding the Degree of Polynomial

The degree of a polynomial is a fundamental concept that determines many of its behaviors. In simple terms, the degree of a polynomial is the highest power of the variable in the polynomial. For the function \( f(x) = x^5 \), the degree is 5 because the variable \( x \) is raised to the fifth power.

Understanding the degree gives us insights about the shape and properties of the polynomial's graph. Here are some key points related to the degree of a polynomial:

• Predicts the Number of Roots: A polynomial of degree \( n \) can have up to \( n \) roots or solutions. For example, a 5th degree polynomial like \( x^5 \) can have up to 5 real roots.
• Determines the Number of Turns: The graph of a polynomial can turn or change direction up to \( n-1 \) times. In our example, \( f(x) = x^5 \) can turn up to four times.
• Influences End Behavior: The degree combined with the leading coefficient tells us how the polynomial behaves as \( x \to \pm \infty \).

Recognizing the degree helps you predict these behaviors and understand the broader storyline the graph is telling.

Recognizing the Leading Coefficient

In a polynomial, the leading coefficient is the coefficient of the term with the highest degree. For the polynomial \( f(x) = x^5 \), the leading coefficient is 1. Although this might seem simple, the leading coefficient plays a significant role in the behavior of the graph.

Here's why the leading coefficient is so important:

• Affects Rising and Falling: If the leading coefficient is positive, the function will eventually rise on both ends, like \( x^5 \). However, if it's negative, the function will fall on both ends.
• Impacts Steepness: The magnitude of the leading coefficient can affect how steep the graph is at large values of \( x \). But since the leading coefficient here is 1, the effect is not pronounced.

This single number helps mathematicians quickly assess how a polynomial's graph will behave without even plotting it. It's like having a sneak peek into the world of the polynomial's limits and is indispensable for understanding graphs.

Performing Interval Analysis

Interval analysis involves examining a function to determine where it increases or decreases over specified ranges of the input variable. In this exercise, we explore the behavior of \( f(x) = x^5 \) over the interval \((-\infty, \infty)\).

Why is interval analysis important?

• Understanding Graph Behavior: This process reveals where the function climbs or falls, which is crucial for sketching the graph by hand.
• Optimization and Real-World Application: In real-world applications, knowing where a function increases can help identify optimal solutions, such as profit maximization.

For \( f(x) = x^5 \), since the leading coefficient is positive, we find that the function increases over the entire interval \((-\infty, \infty)\). This means as \( x \) moves from negative to positive, the function value also continuously rises, giving a clear and predictable trend of the polynomial over any range we consider.