Find all asymptotes, x -intercepts, and y -intercepts for the graph of each rational function an

step 1 To find the asymptotes, we will first consider the point where our function 2 upon x squared plu

Find all asymptotes, x -intercepts, and y -intercepts for...

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Key Concepts

Intercepts

Intercepts are the points where the graph crosses the axes. The x-intercepts occur where the function equals zero (i.e., the numerator is zero provided the denominator is not zero), and the y-intercept is found by evaluating the function at x=0. These key points help locate the graph in relation to the coordinate axes.

Graphing Rational Functions

Graphing a rational function involves plotting its asymptotes and intercepts, and then sketching the curve based on its behavior near the asymptotes and at infinity. This process includes determining the domain, locating any discontinuities, and understanding the function's tendency as x becomes large or approaches points of undefined behavior.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They have unique properties determined by both the numerator and denominator, particularly in understanding their domains, intercepts, and behavior at infinity.

Asymptotes

Asymptotes are lines that a function's graph approaches as the input or output grows large or nears values that are not in the domain. Vertical asymptotes arise when the denominator is zero (and the numerator is nonzero) while horizontal or oblique asymptotes are determined by the degrees of the numerator and denominator polynomials and describe the end behavior of the function.

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