Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

y = 2x/(x + 1)

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Identify the type of rational function: The given function is y = 2x/(x + 1), which is a rational function where the numerator and denominator are polynomials.

Determine the vertical asymptote: Set the denominator equal to zero and solve for x. The vertical asymptote occurs where the function is undefined, which is at x + 1 = 0, so x = -1.

Determine the horizontal asymptote: Compare the degrees of the numerator and denominator. Since both are linear (degree 1), the horizontal asymptote is found by dividing the leading coefficients, which gives y = 2.

Find the intercepts: For the y-intercept, set x = 0 and solve for y, which gives y = 0. For the x-intercept, set y = 0 and solve for x, which gives x = 0.

Analyze the behavior near the asymptotes: As x approaches -1 from the left and right, observe the behavior of the function to understand how it approaches the vertical asymptote. Similarly, as x approaches infinity, observe how the function approaches the horizontal asymptote y = 2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Functions

A rational function is a ratio of two polynomials, typically expressed as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding the behavior of rational functions involves analyzing their domain, asymptotes, and intercepts. The function y = 2x/(x + 1) is a simple rational function where the numerator and denominator are linear polynomials.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For rational functions, vertical asymptotes occur where the denominator is zero, and horizontal asymptotes are determined by the degrees of the polynomials. In y = 2x/(x + 1), the vertical asymptote is x = -1, and the horizontal asymptote is y = 2, as the degrees of the numerator and denominator are equal.

Dominant Terms

Dominant terms in a rational function are those that determine the end behavior of the graph. For y = 2x/(x + 1), the dominant term is 2x/x, which simplifies to 2, indicating the horizontal asymptote. Analyzing dominant terms helps predict how the function behaves as x approaches infinity or negative infinity, crucial for sketching the graph accurately.

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