Oblique Asymptotes


Graph the rational functions in Exercises 103–108. Include the graphs and equations of the asymptotes.


y = x² / (x − 1)

Verified step by step guidance

1. Identify the type of asymptotes: For the given rational function \( y = \frac{x^2}{x - 1} \), we need to find vertical and oblique asymptotes. Vertical asymptotes occur where the denominator is zero, and oblique asymptotes occur when the degree of the numerator is one more than the degree of the denominator.
2. Find the vertical asymptote: Set the denominator equal to zero and solve for \( x \). For \( x - 1 = 0 \), we find \( x = 1 \). Thus, there is a vertical asymptote at \( x = 1 \).
3. Determine the oblique asymptote: Since the degree of the numerator (2) is one more than the degree of the denominator (1), there is an oblique asymptote. Perform polynomial long division of \( x^2 \) by \( x - 1 \) to find the equation of the oblique asymptote.
4. Perform the division: Divide \( x^2 \) by \( x - 1 \). The first term is \( x \), multiply \( x \) by \( x - 1 \) to get \( x^2 - x \). Subtract \( x^2 - x \) from \( x^2 \) to get \( x \). Repeat the process with \( x \) to get the remainder. The quotient \( x + 1 \) is the equation of the oblique asymptote.
5. Graph the function: Plot the vertical asymptote at \( x = 1 \) and the oblique asymptote \( y = x + 1 \). Sketch the curve of the function \( y = \frac{x^2}{x - 1} \) considering the behavior near the asymptotes and the end behavior as \( x \to \pm \infty \).

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding the behavior of rational functions is crucial for graphing them, as they can have vertical, horizontal, or oblique asymptotes depending on the degrees of the polynomials involved.

Oblique Asymptotes

Oblique asymptotes occur in rational functions when the degree of the numerator is exactly one higher than the degree of the denominator. They represent a slant line that the graph approaches as x tends to infinity or negative infinity. To find the equation of an oblique asymptote, perform polynomial long division on the rational function.

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to numerical long division. It is essential for finding oblique asymptotes in rational functions, as it helps determine the linear equation that the graph approaches. The quotient obtained from the division gives the equation of the oblique asymptote.