Question

Find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. $$ G(x)=\frac{x^{3}+1}{x^{2}-5 x-14} $$

Short answer

Vertical asymptotes: x = 7, x = -2. No horizontal asymptote. Oblique asymptote: y = x + 5.

Step-by-step solution

- Finding Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is not zero. Solve the equation of the denominator for zero: \( x^{2} - 5x - 14 = 0 \). Factor the quadratic: \( (x - 7)(x + 2) = 0 \). So, the solutions are \( x = 7 \) and \( x = -2 \). Thus, the vertical asymptotes are at \( x = 7 \) and \( x = -2 \).

- Finding Horizontal Asymptotes

Horizontal asymptotes are determined by comparing the degrees of the numerator and the denominator. The degree of the numerator \( x^{3}+1 \) is 3, and the degree of the denominator \( x^{2}-5x-14 \) is 2. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

- Finding Oblique Asymptotes

Oblique (slant) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. Here, the degree of the numerator is 3 and the degree of the denominator is 2. Perform polynomial long division to divide \( x^{3} + 1 \) by \( x^{2} - 5x - 14 \). Doing so, \( x^{3} ÷ x^{2} = x \), leaving \( x(x^{2} - 5x - 14) = x^{3} - 5x^{2} - 14x \). Subtract this from the original numerator \( x^{3} + 1 -(x^{3} - 5x^{2} - 14x) = 5x^{2} + 14x + 1 \). Continue this process until the degree of the remainder is less than the degree of the denominator. The quotient \( x + 5 \) represents the slant asymptote \( y = x + 5 \).

Key concepts

Vertical Asymptotes

Vertical asymptotes occur at the values of x where the denominator of the rational function equals zero and the numerator is not zero. These asymptotes represent vertical lines on the graph where the function heads towards positive or negative infinity. To find the vertical asymptotes of the function \( G(x)=\frac{x^{3}+1}{x^{2}-5x-14} \), we set the denominator equal to zero and solve for x. The denominator is the quadratic equation \( x^{2} - 5x - 14 \).
Factoring it gives \( (x - 7)(x + 2) = 0 \), which results in the solutions \( x = 7 \) and \( x = -2 \).
Therefore, the vertical asymptotes of the function are at \( x = 7 \) and \( x = -2 \).

When you graph the function, these asymptotes will appear as vertical lines at these x-values, and the function will approach these lines but never touch them.

Horizontal Asymptotes

Horizontal asymptotes illustrate the behavior of the function as \( x \) approaches \( +∞ \) or \( -∞ \). To identify horizontal asymptotes of a rational function, we compare the degrees of the numerator and the denominator:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients of the numerator and the denominator.
• If the degree of the numerator is greater than the degree of the denominator, as in our case with \( G(x) = \frac{x^{3}+1}{x^{2} - 5x - 14} \), there is no horizontal asymptote.

Since the degree of the numerator (3) is greater than that of the denominator (2), the function \( G(x) \) does not possess a horizontal asymptote.

Oblique Asymptotes

Oblique (slant) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. For the function \( G(x) = \frac{x^{3}+1}{x^{2}-5x-14} \), the degree of the numerator is 3 and the degree of the denominator is 2, meeting the condition for an oblique asymptote.
To find the oblique asymptote, polynomial long division is performed. Dividing \( x^{3}+1 \) by \( x^{2}-5x-14 \), results in:

• First, divide the leading terms: \( x^{3} \div x^{2} = x \).
• Multiply \( x \) by \( x^{2} - 5x - 14 \) to get \( x^{3} - 5x^{2} - 14x \).
• Subtract this product from the original polynomial to get the new remainder: \( x^{3} + 1 - (x^{3} - 5x^{2} - 14x) = 5x^{2} + 14x + 1 \).
• Continue this process until the degree of the remainder is less than the degree of the denominator. The quotient is \( x + 5 \), denoting the oblique asymptote \( y = x + 5 \).

The function \( G(x) \) has the oblique asymptote \( y = x + 5 \).

Rational Functions

A rational function is any function that can be expressed as the quotient of two polynomials. For example, \( G(x) = \frac{x^{3}+1}{x^{2}-5x-14} \) is a rational function because both the numerator \( x^{3}+1 \) and the denominator \( x^{2}-5x-14 \) are polynomials.
Rational functions can have vertical, horizontal, or oblique asymptotes, which describe the behavior of the function at certain points or as the independent variable approaches infinity. Understanding the roots of the numerator and the denominator is crucial:

• The vertical asymptotes are found where the denominator is zero, provided the numerator is not zero at those points.
• The horizontal asymptotes depend on the relative degrees of the numerator and the denominator.
• Oblique asymptotes can be identified when the degree of the numerator is exactly one more than the degree of the denominator.

These asymptotes help us grasp the key characteristics of the rational function's graph, such as where it tends to infinity or converges to a line.