Question

Where are the vertical asymptotes of a rational function located?

Short answer

Question: Determine the vertical asymptotes of the rational function \(f(x) = \frac{x^2 - 4}{x^2 - x - 6}\). Answer: To find the vertical asymptotes of the given rational function, follow the steps outlined in the solution. Step 1: Identify the rational function. Here, \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x) = x^2 - 4\) and \(Q(x) = x^2 - x - 6\). Step 2: Find the zeroes of the denominator function (Q(x)). To do this, set Q(x) equal to zero and solve for x: \(x^2 - x - 6 = 0\). Factoring the quadratic equation, we get \((x - 3)(x + 2) = 0\). The zeroes are \(x = 3\) and \(x = -2\). Step 3: Check if the numerator function (P(x)) is not equal to zero at the zeroes of the denominator function. Plug the zeroes of Q(x) into P(x): For \(x = 3\), \(P(3) = (3)^2 - 4 = 5\), which is not equal to zero. For \(x = -2\), \(P(-2) = (-2)^2 - 4 = 0\), which is equal to zero. In this case, we do not have a vertical asymptote at \(x = -2\). Step 4: Write down the vertical asymptotes. The vertical asymptote is located at the x-value where the denominator function is equal to zero and the numerator function is not equal to zero. In this case, the vertical asymptote is at \(x = 3\).

Step-by-step solution

Identify the rational function

Given a rational function, \(f(x) = \frac{P(x)}{Q(x)}\), where P(x) and Q(x) are polynomial functions.

Find the zeroes of the denominator function (Q(x))

Identify the zeroes of Q(x) by determining the x-values that make Q(x) equal to zero. We can do this by setting Q(x) equal to zero and solving for x. Write down the equation: \(Q(x) = 0\)

Check if the numerator function (P(x)) is not equal to zero at the zeroes of the denominator function

Plug the zeroes of Q(x) found in Step 2 into P(x), to ensure that P(x) is not equal to zero at those x-values. If P(x) is not equal to zero, then we have a vertical asymptote at those x-values.

Write down the vertical asymptotes

The vertical asymptotes are located at the x-values found in Step 2, where the denominator function is equal to zero and the numerator function is not equal to zero. Write down all the x-values; these are the vertical asymptotes of the given rational function.

Key concepts

Rational Function Analysis

Understanding how to analyze a rational function is essential for identifying features like vertical asymptotes, which indicate where the function's graph approaches infinity. A rational function is a ratio of two polynomials, mathematically expressed as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions.

Rational function analysis entails several steps. The first is to ensure both P(x) and Q(x) are in their simplest form, often achieved by factoring, to easily find restrictions and important features of the function, such as zeros and asymptotes.

When analyzing these functions, one also looks for domain restrictions – values of x that are not allowed because they would make the function undefined. In such cases, these x-values can cause the graph of the function to have vertical asymptotes. The process of finding vertical asymptotes, as indicated in the textbook solution, involves finding where the denominator of the rational function equals zero, providing those x values don't also zero out the numerator.

Polynomial Zeroes

The concept of polynomial zeroes, also known as roots or solutions, refers to the x-values at which a polynomial function equals zero. These zeroes are crucial in many areas of algebra and calculus, as they help in sketching graphs and solving equations.

To find the zeroes of a polynomial, one often factors the polynomial or uses methods like the quadratic formula or synthetic division for complex polynomials. In the context of a rational function, the zeroes of the denominator polynomial are of special concern. These zeroes can lead to vertical asymptotes, as they point to the x-values where the function is undefined. However, it's vital to check these zeroes against the numerator to ensure they aren't canceled out, which would result in the removal of the potential vertical asymptote.

Discovering and understanding these zeroes empowers students to grasp the broader behavior of polynomials within rational functions and to predict the function's behavior in its graph.

Asymptotic Behavior in Calculus

In calculus, asymptotic behavior is a way of describing how a function behaves near a certain point, or as it approaches infinity. Vertical asymptotes play a pivotal role in this discussion, as they represent the locations where the function grows without bound. More formally, an asymptote is a line that the graph of a function approaches but never actually touches.

The detection of asymptotic behavior becomes a critical part of curve sketching and understanding a function's limits. As 'x' grows larger in either the positive or negative direction, or as we approach a certain x-value, the function might not attain a certain finite value but instead grow indefinitely. This type of behavior is indicative of the limits existing at certain points of a function, and in the case of rational functions, it is often due to zeroes in the denominator that do not coincide with zeroes in the numerator.

Being able to identify and describe asymptotic behavior is fundamental in the study of calculus as it allows students to predict and understand the endless complexity of function behaviors as they relate to real-world phenomena.