Finding Horizontal Asymptotes A Comprehensive Guide

$f(x)=\frac{3-5 x+x^2}{6+7 x+4 x^2}$

Determining the horizontal asymptotes of a rational function is a crucial skill in understanding its behavior, especially as x approaches positive or negative infinity. In this comprehensive guide, we will delve into the process of finding horizontal asymptotes, focusing on the given function:

$f(x)=\frac{3-5 x+x^2}{6+7 x+4 x^2}$

We will explore the underlying principles, step-by-step methods, and provide clear explanations to empower you with the knowledge to confidently tackle similar problems.

Understanding Horizontal Asymptotes

Horizontal asymptotes are horizontal lines that a function approaches as x tends towards infinity (+∞) or negative infinity (-∞). They provide valuable insights into the end behavior of a function, indicating where the function "levels off" as x becomes extremely large or small. To find horizontal asymptotes, we primarily focus on the degrees of the polynomials in the numerator and denominator of a rational function. The degree of a polynomial is the highest power of the variable (x in this case).

For the given function:

$f(x)=\frac{3-5 x+x^2}{6+7 x+4 x^2}$

We observe that the numerator is a polynomial of degree 2 (the highest power of x is $x^2$), and the denominator is also a polynomial of degree 2. This observation is the key to determining the horizontal asymptote(s).

Rules for Finding Horizontal Asymptotes

The relationship between the degrees of the numerator and denominator polynomials dictates the existence and location of horizontal asymptotes. Here's a breakdown of the rules:

1. Degree of Numerator < Degree of Denominator: If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is always the line y = 0 (the x-axis).
2. Degree of Numerator = Degree of Denominator: If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the horizontal line y = (leading coefficient of numerator) / (leading coefficient of denominator). The leading coefficient is the coefficient of the term with the highest power of x.
3. Degree of Numerator > Degree of Denominator: If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, there may be a slant (or oblique) asymptote, which is a diagonal line that the function approaches.

Applying the Rules to Our Function

In our example, we have:

$f(x)=\frac{3-5 x+x^2}{6+7 x+4 x^2}$

The degree of the numerator ($x^2 - 5x + 3$) is 2. The degree of the denominator ($4x^2 + 7x + 6$) is also 2.

Since the degrees are equal, we fall into the second case. Therefore, the horizontal asymptote is given by:

y = (leading coefficient of numerator) / (leading coefficient of denominator)

The leading coefficient of the numerator is 1 (the coefficient of $x^2$). The leading coefficient of the denominator is 4 (the coefficient of $4x^2$).

Thus, the horizontal asymptote is:

y = 1/4

Step-by-Step Solution

Let's summarize the steps we took to find the horizontal asymptote:

1. Identify the degrees of the numerator and denominator polynomials. In our case, both are degree 2.
2. Compare the degrees. Since they are equal, we proceed to the next step.
3. Find the leading coefficients. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 4.
4. Calculate the ratio of the leading coefficients. The ratio is 1/4.
5. Write the equation of the horizontal asymptote. The horizontal asymptote is y = 1/4.

Verification and Interpretation

We have determined that the horizontal asymptote of the function $f(x)=\frac{3-5 x+x^2}{6+7 x+4 x^2}$ is y = 1/4. This means that as x approaches positive or negative infinity, the function's values will get closer and closer to 1/4. The function may cross the horizontal asymptote at some points, but it will generally stay near the line y = 1/4 for very large or very small values of x.

To further verify our result, we can use a graphing calculator or software to plot the function and the line y = 1/4. Observing the graph will visually confirm that the function approaches the horizontal line as x goes to ±∞.

Importance of Horizontal Asymptotes

Understanding horizontal asymptotes is crucial for several reasons:

• End Behavior: They provide key information about how a function behaves for extremely large or small input values.
• Graphing: Horizontal asymptotes help in sketching the graph of a rational function accurately.
• Applications: In real-world applications, horizontal asymptotes can represent limiting values or long-term trends. For example, in population modeling, a horizontal asymptote might indicate the carrying capacity of an environment.

Additional Examples and Practice

To solidify your understanding, let's consider a few more examples:

Example 1:

$g(x)=\frac{2x}{x^2+1}$

The degree of the numerator is 1, and the degree of the denominator is 2. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

Example 2:

$h(x)=\frac{3x^3-x}{x^3+2x^2+1}$

The degree of the numerator and denominator are both 3. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is y = 3/1 = 3.

Example 3:

$k(x)=\frac{x^2+1}{x-2}$

The degree of the numerator is 2, and the degree of the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. There is a slant asymptote in this case, which can be found using polynomial long division.

Conclusion

Finding horizontal asymptotes is a fundamental aspect of analyzing rational functions. By understanding the relationship between the degrees of the numerator and denominator polynomials, we can easily determine the existence and location of these asymptotes. In the case of the function $f(x)=\frac{3-5 x+x^2}{6+7 x+4 x^2}$, we found the horizontal asymptote to be y = 1/4. This skill is not only valuable for mathematical analysis but also for interpreting real-world phenomena modeled by rational functions. Keep practicing with various examples to master this concept and enhance your understanding of function behavior.

This comprehensive guide has equipped you with the knowledge and techniques to confidently find horizontal asymptotes for rational functions. Remember to carefully analyze the degrees of the polynomials and apply the appropriate rules. With practice, you'll become proficient in identifying and interpreting horizontal asymptotes, further strengthening your mathematical prowess.