Problem 17 - Algebra and Trigonometry

Question

Find all horizontal and vertical asymptotes (if any). \(t(x)=\frac{x^{2}}{x^{2}-x-6}\)

Step-by-Step Solution

Answer

Horizontal: \( y = 1 \), Vertical: \( x = 3 \), \( x = -2 \).

Step 1: Define Horizontal Asymptotes

To find horizontal asymptotes, we analyze the degrees of the numerator and the denominator. The function is \( t(x) = \frac{x^{2}}{x^{2} - x - 6} \). Both the numerator and the denominator have a degree of 2. Since their degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients. Therefore, the horizontal asymptote is \( y = \frac{1}{1} = 1 \).

Step 2: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero, provided the numerator is not also zero at those points. To find these, solve \( x^{2} - x - 6 = 0 \). Factoring, we get \((x - 3)(x + 2) = 0\), so \(x = 3\) and \(x = -2\). Thus, the vertical asymptotes are \( x = 3 \) and \( x = -2 \).

Key Concepts

Understanding Rational Functions

Rational functions are expressions that represent the ratio of two polynomials. They take the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomial functions. The polynomial in the numerator, \( P(x) \), and the polynomial in the denominator, \( Q(x) \), determine the behavior of the rational function. Key features of rational functions include:

• Asymptotes, which are lines that the graph of the function approaches but never touches.
• Horizontal asymptotes, which describe the behavior of the graph as \( x \) approaches positive or negative infinity.
• Vertical asymptotes, which indicate where the function is undefined due to division by zero, except when these zeros are canceled by a shared factor in the numerator and denominator.

To analyze a rational function, it's essential to explore its asymptotes, zeros, and any points of discontinuity. This exploration gives insightful information about how the function behaves across different values.

Exploring the Degree of Numerator and Denominator

The degree of a polynomial is the highest power of the variable present. In a rational function, the degrees of the numerator and denominator play a crucial role in determining horizontal asymptotes. Here's how:

• If the degree of the numerator \( P(x) \) is less than the degree of the denominator \( Q(x) \), the horizontal asymptote is \( y = 0 \).
• If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is given by \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of \( P(x) \) and \( Q(x) \), respectively.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

For the given function \( t(x) = \frac{x^2}{x^2 - x - 6} \), both numerator and denominator have a degree of 2. Therefore, the horizontal asymptote is derived from the ratio of their leading coefficients, which are both 1. This results in a horizontal asymptote at \( y = 1 \). Understanding these relationships helps predict how the function behaves at extreme values of \( x \).

Factoring Quadratic Equations

Factoring quadratic equations is a fundamental technique used to solve equations where a quadratic equals zero, such as in finding vertical asymptotes for rational functions. A quadratic equation typically takes the form \( ax^2 + bx + c = 0 \). Factoring involves writing the equation as a product of two binomials, making it easier to identify zero-product scenarios.To factor the quadratic \( x^2 - x - 6 \), we look for two numbers that multiply to \(-6\) (the constant term) and add up to \(-1\) (the coefficient of \(x\)). These numbers are \(-3\) and \(+2\). Thus, the factored form is \((x - 3)(x + 2) = 0\).Setting each factor to zero gives the solutions \(x = 3\) and \(x = -2\). These solutions indicate where the denominator becomes zero, leading to vertical asymptotes at these values of \(x\), because the numerator is non-zero at these points. Knowing how to factor quadratic equations enhances our ability to analyze and understand rational functions deeply.