Question

Identify the horizontal and vertical asymptotes, if any, of the given function. $$ f(x)=\frac{2 x^{2}-2 x-4}{x^{2}+x-20} $$

Short answer

Vertical asymptotes at \(x = -5\) and \(x = 4\); horizontal asymptote at \(y = 2\).

Step-by-step solution

Rewrite the Function

First, we need to rewrite the function in a more manageable form. The given function is \( f(x) = \frac{2x^2 - 2x - 4}{x^2 + x - 20} \). This will help us identify the asymptotes later.

Identify the Vertical Asymptotes

Vertical asymptotes occur where the denominator of a function equals zero, leading to an undefined expression. We find vertical asymptotes by solving \( x^2 + x - 20 = 0 \). Factoring the quadratic, we get \( (x + 5)(x - 4) = 0 \), so the vertical asymptotes are at \( x = -5 \) and \( x = 4 \).

Identify the Horizontal Asymptote

Horizontal asymptotes are determined by comparing the degrees of the polynomial in the numerator and the polynomial in the denominator. Both are degree 2. When the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients. So, \( y = \frac{2}{1} = 2 \). Thus, the horizontal asymptote is \( y = 2 \).

Key concepts

Vertical Asymptotes

Vertical asymptotes are important features of a function that occur where the function is undefined or tends to infinity. For a rational function—where you have a fraction with polynomials in the numerator and the denominator—the vertical asymptotes can be found by identifying the values of the variable that make the denominator equal to zero. This is because division by zero is undefined in mathematics. To find vertical asymptotes, you need to solve for when the denominator equals zero. In our case, take the equation from the denominator \( x^2 + x - 20 = 0 \). Solving this by factorization, \( (x + 5)(x - 4) = 0 \), we get the solutions \( x = -5 \) and \( x = 4 \). These are the x-values where vertical asymptotes are located, which means the graph of the function will approach these lines but never actually touch them.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input variable approaches infinity or negative infinity. For rational functions, the horizontal asymptote depends on the degrees of the polynomials in the numerator and denominator. The degree of a polynomial is the highest power of the variable in the expression. If the degrees are equal, like in our function where both numerator and denominator are degree 2, look at the leading coefficients (the coefficients of the terms with the highest power). The horizontal asymptote is the ratio of these coefficients. In our example, the leading coefficients are 2 for the numerator and 1 for the denominator, meaning the horizontal asymptote is \( y = \frac{2}{1} = 2 \). This indicates that as \( x \) goes very large in positive or negative direction, the function \( f(x) \) will approach the value 2.

Polynomial Factorization

Polynomial factorization is a mathematical process used to rewrite a polynomial expression as a product of its factors. Factoring simplifies solving equations, especially in finding asymptotes, as seen in this exercise. To factor a polynomial, you need to express it as a product of simpler polynomials, which when multiplied together give the original polynomial. Often, this involves finding the roots or solutions where the polynomial equals zero. In our example, the denominator \( x^2 + x - 20 \) was factored to \( (x + 5)(x - 4) \). Each factor represents a potential vertical asymptote because they individually equal zero at their respective roots \( x = -5 \) and \( x = 4 \). Factoring is especially useful for functions because it helps us understand their behavior more thoroughly.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. They are an essential type of function in calculus and algebra, providing insights into real-world phenomena and mathematical theory. These functions are expressed in the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. Rational functions exhibit unique characteristics such as vertical and horizontal asymptotes, which describe how the functions behave for extremely large or small values of \( x \). In essence, analyzing a rational function involves checking the degrees of its numerator and denominator, factorizing to find roots, and determining its asymptotes. These steps help us understand how the function changes its direction, grows, or levels off, making them particularly valuable in graphing and in predicting behaviors in applied contexts.