Graphing Simple Rational Functions

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

y = −3/(x − 3)

Verified step by step guidance

Identify the type of rational function: The given function is y = -3/(x - 3), which is a simple rational function with a single term in the denominator.

Determine the vertical asymptote: Set the denominator equal to zero, x - 3 = 0, which gives x = 3. This is the vertical asymptote of the function.

Determine the horizontal asymptote: Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

Analyze the behavior near the asymptotes: As x approaches 3 from the left, the function y = -3/(x - 3) tends to negative infinity, and as x approaches 3 from the right, the function tends to positive infinity.

Sketch the graph: Plot the vertical asymptote at x = 3 and the horizontal asymptote at y = 0. Draw the curve approaching these asymptotes, showing the behavior described in the previous step.

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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 = -3/(x - 3) is a simple rational function with a linear polynomial in the denominator.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For rational functions, vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe end behavior. In y = -3/(x - 3), the vertical asymptote is x = 3, indicating where the function is undefined and the graph approaches infinity.

Dominant Terms

Dominant terms in a rational function determine its behavior as x approaches infinity or negative infinity. For y = -3/(x - 3), the dominant term is -3/x, which influences the horizontal asymptote. As x becomes very large or very small, the function approaches y = 0, indicating a horizontal asymptote at y = 0.

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