In Exercises 51-54, graph each equation. y=(1)/(x)( Let x=-2,-1,-(1)/(2),-(1)/(3), (1)/(3), (1)

step 1 want to graph y is equal to 1 divided by x and to do so what we can do is plug in different valu

In Exercises 51-54, graph each equation. y=(1)/(x)( Let ...

Key Concepts

Rational Functions

Rational functions are defined as the quotient of two polynomials, and they often exhibit interesting features such as asymptotes, discontinuities, and distinct branches. The behavior of these functions can vary significantly depending on the zeros and degrees of the numerator and denominator, which makes graphing an essential tool for visualizing their properties.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. In the context of rational functions, vertical asymptotes occur where the denominator is zero, indicating points of discontinuity, while horizontal asymptotes describe the end behavior of the function at infinity, often determined by the degrees of the numerator and denominator.

Hyperbola

A hyperbola is one of the conic sections characterized by two disconnected curves (branches) that mirror each other with respect to the origin in the case of a simple reciprocal function. The graph of a reciprocal function typically forms a hyperbola, demonstrating the inverse relationship between the variables.

Symmetry

Symmetry in a function’s graph can greatly simplify the process of graphing. For example, if a function is odd, its graph is symmetric about the origin; this means that if the point (x, y) is on the graph, then the point (-x, -y) will also be on the graph. Recognizing symmetry assists in sketching accurate graphs without computing every point.

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