Graph each rational function. Give the equations of the vertical and horizontal asymptotes. See

step 1 Let's create a graph of f of x equals 2 over x. In order to create this graph, we're going to lo

Graph each rational function. Give the equations of the...

Key Concepts

Rational Functions

A rational function is a function expressed as the ratio of two polynomials. The properties of these functions, such as discontinuities, intercepts, and asymptotic behaviors, are derived from their numerator and denominator. They are a central object of study in algebra and calculus for analyzing behavior of functions across their domains.

Vertical Asymptotes

Vertical asymptotes occur in rational functions at the values of x that make the denominator zero, assuming these values do not also cancel with factors in the numerator. At these points, the function tends to infinity or negative infinity, indicating a type of discontinuity where the function is undefined.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches positive or negative infinity. In rational functions, these are determined by comparing the degrees of the numerator and denominator polynomials. They represent the limiting value of the function as the input grows large, and are crucial in understanding the end behavior of the function.

Graphing Rational Functions

Graphing a rational function involves identifying key features such as the x and y intercepts, and the vertical and horizontal asymptotes, as well as any holes. This process provides a skeletal framework that guides the accurate sketching of the function's behavior across its domain and underlines the significance of asymptotic behavior in the overall graph.

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