Problem 3 - College Algebra with Corequisite Support

Question

In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.

Step-by-Step Solution

Answer

The function rises to infinity on the right and falls to negative infinity on the left.

Step 1: Understand the Power Function

A power function is generally given by the form \( f(x) = ax^n \), where \( a \) is the leading coefficient and \( n \) is the degree of the function. If \( n \) is odd, it means \( n = 1, 3, 5, \ldots \)

Step 2: Analyze Behavior for Positive Leading Coefficient

When the leading coefficient \( a \) is positive, as \( x \to \infty \) (positive infinity), \( ax^n \to \infty \), and as \( x \to -fty \) (negative infinity), \( ax^n \to -fty \).

Step 3: Draw a Conclusion on End Behavior

For a power function \( f(x) = ax^n \) with an odd degree \( n \) and positive leading coefficient \( a \), the graph approaches \( fty \) as \( x \to fty \) and approaches \( -fty \) as \( x \to -fty \). This results in a tail moving up on the right end of the graph and down on the left end.

Key Concepts

Positive leading coefficient

In a power function, the leading coefficient plays a crucial role in determining the behavior of the graph. When we have a positive leading coefficient, it means the coefficient of the term with the highest power (degree) is greater than zero.
This directly affects how the function behaves at the far ends of the graph.
For an odd degree power function, a positive leading coefficient influences the direction of the tails of the graph:

• As the value of the input variable increases to positive infinity, the output (or y-value) also climbs to positive infinity.
• Conversely, as the input heads towards negative infinity, the output drops towards negative infinity.

Thus, the nature of the positive leading coefficient ensures that one end of the graph will always rise as it extends towards infinity, impacting the function's end behavior in a predictable way.

Odd degree power functions

Odd degree power functions are functions where the highest power of the variable is an odd number, such as 1, 3, 5, and so forth.
These functions have distinct characteristics that set them apart from even degree power functions.
Because of their odd nature, these functions exhibit a different pattern when analyzed for end behavior, especially when combined with a positive leading coefficient.

• Odd degree functions will always cross the x-axis at least once, which means they have at least one real root.
• The simple linear function, which is a special case of odd degree functions ( n=1 ), appears as a straight line.

Understanding these characteristics helps in predicting how such functions will behave without needing to plot them.

Graph behavior analysis

Analyzing the graph behavior of odd degree power functions involves observing how the function behaves as the input values become extremely large or small, known as end behavior. With odd degree power functions, especially those with a positive leading coefficient, certain graphical trends become apparent:

• The graph rises to the top right as x approaches positive infinity.
• It falls to the bottom left as x goes towards negative infinity.

This pattern of rising and falling resembles an 'S' shape when drawn, showing how the function stretches over the x-axis.
This analysis is crucial for understanding the function's full scope, including how it transitions smoothly across various sections of the graph.

Power function properties

Power functions have specific properties based on their degree and leading coefficient, which help determine their graph's end behavior and shape.
These properties include:

• The degree of the function: Determines how many roots or x-intercepts the function might have.
• The leading coefficient: Determines the direction of the tails if the degree is odd, or the parabola's "bowl" shape if the degree is even.
• The symmetry of the graph: Odd degree functions show rotational symmetry around the origin, implying they look similar after a 180-degree rotation.
• Monotonic behavior: Odd degree power functions may not maintain constant increasing or decreasing patterns but will eventually rise or fall predictably.

Recognizing these properties can make understanding and graphing power functions less complex, providing a clearer insight into their fundamental behaviors.