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Delving into the realm where** patterns**,** functions**, and **behaviors** take the **forefront**, we explore how to find **end behavior** in mathematics. An intriguing notion is ‘end behavior,’ deeply ingrained in mathematical analysis and **calculus**.

This term provides us with a window into the future trajectory of a function, depicting the path it will take as its inputs inch ever closer toward the extremes of **infinity**.

The article will explore the concept in depth, spotlight its practical applications, and demonstrate how it is a potent tool for **mathematicians**, **engineers**, and **scientists**.

**How to Find the E****nd Behavior**

**nd Behavior**

To **find the end behavior** of a **polynomial function**, examine the** highest-degree** term. The sign and degree of this term will determine the behavior of the function as **$x$** approaches positive or negative infinity.

In mathematics, ‘**end behavior**‘ refers to the values a function approaches as its input (or the independent variable) heads toward positive or negative **infinity**. It provides insights into how a function behaves in its domain’s extremes or ends.

This behavior is particularly vital in studying **limits**, **asymptotes**, and **infinite behavior** of functions. Typically described using limit notation, the **end behavior** of a function can convey its growth or decay patterns and how it behaves **‘at the ends,’** giving us a crucial perspective on the function’s overall behavior and potential **practical applications**.

## Understanding the End Behavior

Understanding **end behavior** in mathematics is about grasping how a function behaves as its input (often denoted as **x**) approaches positive or negative **infinity**. It is essentially a way to describe a function’s long-term **behavior** or** trends**. In simpler terms, it tells us what happens to a function’s output (or **y-values**) as the input becomes very large (either positively or negatively).

The **end behavior** of a function is primarily determined by its highest **degree** term (in **polynomial functions**) or by the ratio of the degrees of the numerator and denominator (in **rational functions**). Here are some rules that can help in understanding the **end behavior** of different types of functions:

**Polynomial Functions**

If the **degree** of the polynomial is even, then the function’s ends will either point up or both points down, depending on the sign of the **leading coefficient**. If the **degree** is odd, then if the **leading coefficient** is positive, the function will start low (as **x** approaches negative **infinity**) and end high (as **x** approaches positive **infinity**). If the **leading coefficient** is negative, the function will start high and end low. Below we present a generic polynomial function in Figure-1.

Figure-1. Generic polynomial function.

**Rational Functions**

If the **degree** of the numerator is less than the **degree** of the denominator, the function approaches 0 as **x** approaches positive or negative **infinity**. If the degrees are equal, the **end behavior** is the ratio of the **leading coefficients**. If the **degree** of the numerator is greater than the **degree** of the denominator, the function approaches positive or negative **infinity** as **x** approaches positive or negative **infinity**, depending on the coefficients’ signs. Below we present a generic rational function in Figure-2.

Figure-2. Generic rational function.

**Exponential Functions**

For **exponential functions**, if the base is greater than 1, the function approaches **infinity** as **x** approaches **infinity** and 0 as **x** approaches negative **infinity**. If the base is a fraction between 0 and 1, the function approaches 0 as **x** approaches **infinity** and **infinity** as **x** approaches negative **infinity**. Below we present a generic exponential function in Figure-3.

Figure-3. Generic exponential function.

Understanding the **end behavior** of a function is an important concept in **calculus** and many other branches of mathematics, and it has numerous real-world applications in fields such as **physics**, **economics**, and **computer science**.

**Process of How to Find ****End Behavior **

Finding the **end behavior** of a function typically involves analyzing its **degree** and **leading coefficient**. This is commonly done with **polynomial functions**, but the concept can apply to other functions. Here is a general process:

**Identify the Type of Function**

It’s important to recognize the type of function you’re working with, as different functions have different methods to find their **end behavior**. For **polynomials**, you’ll look at the highest power term (**degree**) and its **leading coefficient**.

**Determine the Degree of the Function**

For **polynomial functions**, the **degree** is the highest power of the variable within the function. The **degree** of the function can tell us whether the function ends up or down as we read from left to right.

**Identify the Leading Coefficient**

Correct, the **leading coefficient** is the term’s coefficient with the highest degree in a polynomial function. The **leading coefficient** can tell us whether the function is positive or negative as we move toward infinity.

**Analyze the End Behavior**

Based on the **degree** and** leading coefficient**, we can make the following conclusions:

- If the
**degree**is**even**, and the**leading coefficient**is positive, the end behavior is: as**x**approaches positive or negative infinity,**y**approaches positive infinity. In simple terms, both ends of the graph**point upwards**. - If the degree is even, and the leading coefficient is
**negative**, as x approaches positive or negative infinity, y approaches**negative infinity**. Both ends of the graph point**downwards**. - If the degree is
**odd**, and the leading coefficient is**positive**,**x**approaches**negative infinity**,**y**approaches**negative infinity**, and as**x**approaches**positive infinity**,**y**approaches**positive infinity**. The graph**falls**to the left and**rises**to the right. - If the degree is
**odd**, and the leading coefficient is**negative**,**x**approaches**negative infinity**,**y**approaches**positive infinity**, and as**x**approaches**positive infinity**,**y**approaches**negative infinity**. The graph**rises**to the left and**falls**to the right.

It’s important to note that these rules apply to **polynomial functions**. Different rules or techniques may be needed to determine end behavior for other functions, such as **rational, exponential, or logarithmic functions**.

**Properties**

Understanding the **end behavior** of a function provides insights into its behavior as it approaches infinity in the positive or negative direction. Here are some essential properties of end behavior that are crucial for **analysis**:

**End Behavior of Polynomial Functions**

As mentioned earlier, the end behavior of **polynomial functions** is determined by the function’s **degree** and **leading coefficient**. If the degree is **even**, the end behavior of the function will be the same in both directions (both arms of the graph either point upwards or downwards). If the degree is **odd**, the end behavior of the function will be different in both directions (one arm of the graph **points upwards**, and the other **points downwards**).

**End Behavior of Rational Functions**

A **rational function** is a function that can be expressed as a fraction of two polynomials. The end behavior of a rational function depends on the degrees of the **numerator** and **denominator polynomials**.

- If the
**degree**of the**numerator**is larger, the function approaches positive or negative infinity as**x**approaches positive or negative infinity. - If the
**degrees**of the**numerator**and denominator are the same, the function approaches the**ratio**of the**leading coefficients**of the numerator and denominator. - If the
**degree**of the d**enominator**is larger, the function approaches**0**as**x**approaches positive or negative infinity.

**End Behavior of Exponential Functions**

For **exponential functions**, the end behavior depends on whether the **base** is greater than one or between zero and one.

- If the base is
**greater than one**, the function approaches**infinity**as x approaches**infinity**and**zero**as x approaches**negative infinity**. - Conversely, if the base is
**between zero and one**, the function approaches**zero**as x approaches**infinity**and approaches**infinity**as x approaches**negative infinity**.

**End Behavior of Logarithmic Functions**

For **logarithmic functions**, as x approaches **positive infinity**, the function also approaches **positive infinity**. However, the function approaches **negative infinity** as x approaches **zero** from the right.

**End Behavior of Trigonometric Functions**

**Trigonometric functions** like **sine** and **cosine** do not have end behaviors in the conventional sense. These functions **oscillate** between fixed values and do not approach **infinity** or **negative infinity** as x increases or decreases. They exhibit periodic behavior instead of approaching specific values at the ends of the graph.

**End Behavior and Limits**

The concept of **limits** is heavily tied to **end behavior**. The **end behavior** is often described using **limit notation**, which precisely describes the behavior of a function as it approaches a particular value or **infinity**.

**End Behavior and Asymptotes**

**Horizontal** and **slant asymptotes** describe the **end behavior** of a function. An **asymptote** is a line that the function approaches but never quite reaches. The existence and direction of **asymptotes** can provide valuable insights into the function’s **end behavior**.

These properties of **end behavior** serve as crucial analytical tools to understand the behavior of functions towards the ends of their domains, guiding mathematical, engineering, or scientific problem-solving.

## Significance

Understanding the end behavior of functions in **mathematics** is critical for several reasons:

**Predicting Long-term Trends**

The **end behavior** of a function helps us understand what happens to the function as the input values get very large or very small, in other words, what happens “in the long run”. This is particularly useful in fields such as **physics**, **economics**, or any area where modeling and prediction over extended periods or large ranges is required.

**Analyzing the Behavior of Complex Functions**

Often, **complex functions** are difficult to analyze due to their structure. Studying the **end behavior** can provide valuable insight into the overall behavior of the function, aiding in its understanding and interpretation.

**Helping Determine Function Type**

The **end behavior** can also provide clues about the type of function. For instance, even-degree polynomials have the same **end behavior** at positive and negative infinity, whereas odd-degree polynomials have different **end behavior** at positive and negative infinity.

**Assessing Function Asymptotes**

In rational functions, by comparing the degrees of the polynomial in the numerator and the denominator, we can predict the **end behavior**, which in turn helps us identify **horizontal or slant asymptotes**.

**Comparing and Classifying Functions**

The study of **end behavior** allows us to compare different **functions** and classify them according to their behavior as the **input** approaches **infinity**. This is a fundamental part of the study of **algorithmic complexity** in **computer science**, where functions are classified based on how their **runtime** grows as the size of the input increases.

**Limit Calculations**

**End behavior** is directly related to **limits at infinity**, an important concept in **calculus**. This is key to understanding concepts like **continuity**, **differentiability**, **integrals**, and **series**.

By understanding **end behavior**, mathematicians and scientists can better understand the characteristics of different functions and apply this knowledge to solve complex problems and make predictions.

**Limitations of End Behavior**

While the concept of end behavior is a powerful tool in **mathematical analysis**, it does come with its set of limitations:

**Not All Functions Have Defined End Behavior**

Some functions, like **periodic functions** (sine and cosine), do not have an **end behavior** in the traditional sense as they **oscillate** between two fixed values and never approach positive or negative **infinity**.

**Inapplicable for Discontinuous Functions**

For functions that are **discontinuous** or **undefined** at some points, the concept of **end behavior** might not provide a clear understanding of the function’s behavior.

**Limitations With Complex Functions**

When dealing with **complex functions**, determining **end behavior** can be more challenging as these functions might have different behaviors in different directions approaching **infinity**.

**Lack of Information on Local Behavior**

The **end behavior** gives us insights into the behavior of a function as it approaches positive or negative **infinity**. Still, it tells us little about what happens in the middle, also known as the **local behavior** of the function. Thus, it cannot be used as the sole tool to understand a function completely.

**Infinite Oscillations**

In some cases, functions can **oscillate** infinitely as they approach a limit, making it difficult to discern a clear **end behavior**. An example is the function **f(x) = sin(1/x)** as **x** approaches **0**.

**Inability to Handle Ambiguity**

In certain situations, the **end behavior** of a function may be **ambiguous** or **undefined**. For instance, the function **1/x²** oscillates between positive and negative infinity as **x** approaches **0**.

Thus, while **end behavior** is an important tool for understanding how functions behave as they approach infinity, it is not a universal solution. It must be used with other analytical tools to provide a more comprehensive understanding of a function.

**Applications **

The concept of **end behavior** in **mathematics** has numerous applications in various fields and real life. By examining the **end behavior**, we can better understand various **phenomena**. Here are some examples:

**Physics and Engineering**

In **physics**, **end behavior** can be used to model and predict the behavior of physical systems. For instance, an engineer designing a bridge might use **polynomial functions** to model the stresses on different bridge parts. Understanding the **end behavior** of these functions can help predict what will happen under extreme conditions, like high winds or heavy loads.

**Economics and Finance**

In economics, **end behavior** is often used to create models to predict future trends. Economists can use functions to model data like **inflation rates**, **economic growth**, or **stock market trends**. The **end behavior** of these functions can indicate whether the model predicts ongoing growth, eventual stagnation, or cyclical behavior.

**Environmental Science**

In environmental science, **end behavior** can be used to predict the outcome of certain phenomena. For example, a model might use a function to represent the **population growth** of a species. The **end behavior** of this function can give insights into whether the population will eventually stabilize, continue growing indefinitely, or oscillate in size.

**Computer Science**

In computer science, especially in algorithm analysis, **end behavior** is used to describe the **time complexity** of an algorithm. By examining the **end behavior** of a function representing the algorithm’s runtime, one can infer how the algorithm will perform as the input size approaches infinity.

**Real-life scenarios**

In real life, understanding **end behavior** can help predict various phenomena. For example, a business owner might use a function to model their **sales** over time. By studying the **end behavior**, they can predict whether their sales will **increase**, **decrease**, or **stay the same** long-term.

**Medicine and Pharmacology**

**End behavior** is crucial in modeling the rate at which a drug is **metabolized** in the body or how the concentration of a medication changes over time in the **bloodstream**. As such, understanding the **end behavior** of the relevant functions can help physicians determine the right dosage and frequency of medication for patients.

**Meteorology**

In meteorology, functions may be used to model **weather patterns** or **atmospheric conditions** over time. The **end behavior** of these functions can provide insights into long-term **climate trends** or potential **extreme weather events**.

**Population Dynamics**

In biology and ecology, **end behavior** is used in **population dynamics** models. By understanding the **end behavior** of these models, scientists can predict whether a species’ **population** will **grow indefinitely**, **stabilize**, or eventually become **extinct**. This is particularly useful in **conservation efforts** for **endangered species**.

**Astrophysics**

The concept of **end behavior** is also used in **astrophysics**. For example, functions may describe a star’s **lifecycle** or the universe’s **expansion**. The **end behavior** of these functions provides insights into the future state of these celestial objects or systems.

**Market Research**

Companies use **end behavior** to forecast past sales or market data trends. It helps them in **strategic planning**, like when to launch new products, enter new markets, or phase out old services.

**Agriculture**

Farmers and agricultural scientists use models that involve **end behavior** to predict crop yields based on various factors such as **rainfall**, **fertilizer use**, and **pest infestations**. Understanding these models’ **end behavior** can help develop strategies for increasing **productivity** and **sustainability**.

In all these fields and more, understanding the **end behavior** of functions provides critical insights and helps make informed **predictions** and **decisions**.

**Exercise **

### Example 1

**Polynomial Function**

Find the end behavior of the function: **f(x) = 2x⁴ – 5x² + 1**

Figure-4.

### Solution

The highest degree (4) is even, and the leading coefficient (2) is positive. Therefore, as x approaches positive or negative infinity, f(x) also approaches positive infinity. In terms of notation, we write this as:

lim (x->+∞) f(x) = +∞

lim (x->-∞) f(x) = +∞

### Example 2

#### Polynomial Function

Find the end behavior of the function: **f(x) = -3x^5 + 4x³ – x + 2**

### Solution

The highest degree (5) is odd, and the leading coefficient (-3) is negative. Therefore, as x approaches positive infinity, f(x) approaches negative infinity, and as x approaches negative infinity, f(x) approaches positive infinity. We write this as:

lim (x->+∞) f(x) = -∞

lim (x->-∞) f(x) = +∞

**Example 3**

**Rational Function**

Find the end behavior of the function: **f(x) = (3x² + 2) / (x – 1)**

Here, the degree of the numerator (2) is higher than that of the denominator (1). Thus, as x approaches positive or negative infinity, f(x) also approaches positive or negative infinity, depending on the sign of x. We write this as:

lim (x->+∞) f(x) = +∞

lim (x->-∞) f(x) = -∞

### Example 4

**Rational Function**

Find the end behavior of the function: **f(x) = (2x + 1) / (x² – 4)**

### Solution

Here, the degree of the numerator (1) is less than that of the denominator (2). Therefore, as x approaches positive or negative infinity, f(x) approaches 0. We write this as:

lim (x->+∞) f(x) = 0

lim (x->-∞) f(x) = 0

**Example 5**

**Exponential Function**

Find the end behavior of the function: **f(x) = 2ᵡ**

### Solution

As x approaches positive infinity, f(x) approaches positive infinity. And as x approaches negative infinity, f(x) approaches 0. We write this as:

lim (x->+∞) f(x) = +∞

lim (x->-∞) f(x) = 0

**Example 6**

**Cubic Function**

Find the end behavior of the function: **f(x) = 3x³**

Figure-5.

### Solution

The degree is 3, which is odd, and the leading coefficient (3) is positive. Therefore, as x approaches positive infinity, f(x) also approaches positive infinity, and as x approaches negative infinity, f(x) approaches negative infinity. We write this as:

lim (x->+∞) f(x) = +∞

lim (x->-∞) f(x) = -∞

This end behavior is typical for cubic functions with a positive leading coefficient. As x gets large in either the positive or negative direction, the term with the highest power (3) dominates the function, leading to the observed end behavior.

**Example 7**

**Quadratic Function**

Find the end behavior of the function: **f(x) = -2x² + 3x + 1**

The highest degree is 2, which is even, and the leading coefficient (-2) is negative. Therefore, as x approaches positive or negative infinity, f(x) approaches negative infinity. We write this as:

lim (x->+∞) f(x) = -∞

lim (x->-∞) f(x) = -∞

Quadratic functions with a negative leading coefficient always decrease towards negative infinity as x gets large in either the positive or negative direction.

### Example 8

**Exponential Function**

Find the end behavior of the function: **f(x) = $\left(\frac{1}{3}\right)^{x}$**

Here, the base is less than one. Thus, as x approaches positive infinity, f(x) approaches 0. And as x approaches negative infinity, f(x) approaches positive infinity. We write this as:

lim (x->+∞) f(x) = 0

lim (x->-∞) f(x) = +∞

*All images were created with MATLAB.*