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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10m

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Graphing Simple Rational FunctionsGraph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.y = 2x/(x + 1)