In Exercises 57-62 , find the intercepts, vertical...

In Exercises $57-62$ , find the intercepts, vertical asymptotes, and end behavior asymptote, and graph the function together with its end behavior asymptote.
$$
h(x)=\frac{2 x^{3}-3 x+2}{x^{3}-1}
$$

Key Concepts

End Behavior Asymptote

The end behavior asymptote describes how a function behaves as x approaches positive or negative infinity. For rational functions, this is often determined by comparing the degrees of the numerator and the denominator, using limits or polynomial long division. This asymptote could be horizontal, oblique, or even another type, depending on the degree relationship, and it provides insight into the function's long-term behavior.

Graphing Rational Functions

Graphing rational functions involves combining information from intercepts, vertical asymptotes, and end behavior asymptotes. By plotting these key features, one can sketch an accurate representation of the function. Additional local behavior such as holes in the graph (removable discontinuities) may also be considered to enhance the precision of the graph.

Intercepts

Intercepts are the points where the graph of a function crosses the axes. The x-intercepts are found by setting the function equal to zero and solving for x, while the y-intercept is determined by evaluating the function at x = 0. These points provide a starting framework for graphing and understanding the behavior of the function in relation to the coordinate axes.

Vertical Asymptotes

Vertical asymptotes occur at values of x where the function becomes undefined, typically due to division by zero after canceling any common factors between the numerator and denominator. They indicate the values at which the graph increases or decreases without bound, and they are critical for understanding the function's domain and local behavior near these points.

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