What Is End Behavior Of A Function

Introduction

In mathematics, end behavior is the overall shape of a graph of a function as it approaches infinity or negative infinity. The end behavior can be determined by looking at the leading term of the function. The leading term is the term with the largest exponent in a polynomial function. For example, in the polynomial function f(x) = 3×4 + 2×3 – 5×2 – 6x + 2, the leading term is 3×4. The end behavior of a function is affected by the sign of the leading term. If the leading term is positive, then the graph of the function will approach infinity as x goes to infinity. If the leading term is negative, then the graph of the function will approach negative infinity as x goes to infinity.

What is End Behavior?

End behavior can be thought of as what happens to a graph of a function as x approaches positive or negative infinity. In other words, it’s the long-term trend of a function.

There are two types of end behavior:

1) Asymptotic: this is when a graph approaches a certain value (or values) as x approaches infinity. For example, the graph of y = 1/x becomes closer and closer to the y-axis as x approaches infinity.

2) Unbounded: this is when a graph doesn’t approach any specific value, but just keeps growing (or shrinking) without bound. An example of this is the exponential function y = 2^x – this function gets larger and larger without ever approaching any specific value (other than infinity).

Factors That Affect End Behavior

There are a few key factors that affect the end behavior of a function:

-The degree of the polynomial function: The higher the degree, the more extreme the end behavior will be. For example, a cubic function will have much sharper peaks and valleys than a quadratic function.

-The sign of the leading coefficient: This determines whether the ends will approach positive or negative infinity. If the leading coefficient is positive, then the ends will approach positive infinity; if it is negative, then the ends will approach negative infinity.

-The behavior of the other coefficients: If all of the other coefficients are positive, then both ends will be concave up; if all of them are negative, then both ends will be concave down. However, if some coefficients are positive and others are negative, then one end will be concave up while the other end is concave down.

How to Determine End Behavior

When graphing a polynomial function, it is important to be able to identify the end behavior. The end behavior of a polynomial function is determined by the leading term. The leading term is the term with the highest exponent in the function. For example, in the function ƒ(x) = 3×4 + 2×3 – 5×2 + 7, the leading term is 3×4. To determine the end behavior of ƒ(x), we need to look at the sign of the leading term’s exponent and coefficient. In this case, both are positive, so we know that ƒ(x) will have a positive lead coefficient and will approach infinity as x approaches infinity. Similarly, if both the exponent and coefficient were negative, we would know that ƒ(x) would have a negative lead coefficient and would approach negative infinity as x approaches infinity. However, if one is positive and one is negative (like in our example), then we need to further examine the function to determine its end behavior.

Examples of End Behavior

End behavior of a function is the way the graph of the function “ends” as x approaches positive infinity or negative infinity. In other words, it’s what happens to the y-value of a graph as x gets extremely large or extremely small.

There are two types of end behavior: left end behavior and right end behavior. Left end behavior is what happens when x approaches negative infinity, while right end behavior is what happens when x approaches positive infinity.

For example, consider the graph of y = 1/x:

As x approaches negative infinity, the graph approaches 0 from below. This is called left-hand limit behavior. As x approaches positive infinity, the graph also approaches 0 but from above. This is called right-hand limit behavior.

Conclusion

End behavior of a function refers to how the graph of the function behaves as x approaches positive or negative infinity. In other words, it describes what the graph looks like at the “ends” of the x-axis. The end behavior can be used to identify the overall shape of a graph, even if there is no apparent pattern in the data. It is a helpful tool for analyzing functions and can be used to predict how a function will behave over certain intervals.