Behavior at Infinity

Definition

Behavior at infinity refers to how a function behaves as the input values approach positive or negative infinity. In the context of polynomial functions, this term is essential for understanding the end behavior of their graphs, including whether they rise or fall as x approaches infinity or negative infinity. Analyzing this behavior can also provide insights into the degree of the polynomial and its leading coefficient, which significantly influence the graph's overall shape.

5 Must Know Facts For Your Next Test

1. The end behavior of a polynomial function is determined by its leading term, specifically its degree and leading coefficient.
2. For even-degree polynomials with a positive leading coefficient, both ends of the graph will rise to positive infinity; for even-degree polynomials with a negative leading coefficient, both ends will fall to negative infinity.
3. Odd-degree polynomials with a positive leading coefficient will rise to positive infinity on the right and fall to negative infinity on the left; odd-degree polynomials with a negative leading coefficient will do the opposite.
4. As x approaches infinity or negative infinity, polynomial functions will dominate over lower-degree terms, making them less significant for determining end behavior.
5. Understanding behavior at infinity helps predict how a polynomial function will behave outside its local maximum and minimum points.

Review Questions

• How does the leading coefficient influence the behavior at infinity for polynomial functions?
  • The leading coefficient plays a crucial role in determining how a polynomial function behaves at infinity. If the leading coefficient is positive, an even-degree polynomial will rise on both ends, while an odd-degree polynomial will rise on the right and fall on the left. Conversely, if the leading coefficient is negative, an even-degree polynomial will fall on both ends, while an odd-degree polynomial will fall on the right and rise on the left. Therefore, knowing whether the leading coefficient is positive or negative helps predict end behavior.
• Explain how you would determine the end behavior of a polynomial function based on its degree.
  • To determine the end behavior of a polynomial function based on its degree, first identify whether the degree is even or odd. For even degrees, if the leading coefficient is positive, both ends of the graph will rise; if negative, both ends will fall. For odd degrees, a positive leading coefficient means one end rises while the other falls; if negative, it reverses. This understanding allows you to sketch accurate end behavior without needing specific function values.
• Analyze how understanding behavior at infinity can aid in sketching graphs of complex polynomial functions.
  • Understanding behavior at infinity equips you with essential insights into how complex polynomial functions behave beyond their critical points. By recognizing whether a function rises or falls at both ends based on its degree and leading coefficient, you can create accurate sketches that reflect overall trends. This approach allows you to see where turning points may occur and provides context for local maxima and minima within the graph. Ultimately, this knowledge streamlines your graphing process and enhances your comprehension of polynomials as a whole.

Related terms

End Behavior: End behavior describes the direction in which a graph heads as the input values increase or decrease indefinitely.

Leading Coefficient: The leading coefficient is the coefficient of the term with the highest degree in a polynomial, determining the direction of the graph at infinity.

Degree of a Polynomial: The degree of a polynomial is the highest power of the variable in the polynomial expression, which influences the number of turning points and overall shape of its graph.