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When graphing rational functions, vertical asymptotes play a crucial role. They occur at points where the function is undefined, leading to a dramatic change in direction on the graph. To identify vertical asymptotes, you need to set the denominator equal to zero. For example, in the function \( y = \frac{x^2}{x-1} \), setting the denominator \( x-1 \) equal to zero gives us \( x = 1 \).
This means the graph will approach but never touch the vertical line \( x = 1 \). As \( x \) gets close to 1 from the left or right, the function's values will increase or decrease without bound (towards infinity).
This is why it seems like the graph is splitting at this point, creating a line of discontinuity known as a vertical asymptote. Always look for points where the denominator reaches zero to find these asymptotes.

Horizontal asymptotes differ from vertical ones because they concern the end behavior of a graph, not any particular x-value where the function is undefined. They exist in rational functions when the degree of the numerator is less than or equal to the degree of the denominator.
To find a horizontal asymptote, you compare these degrees. In our example, \( y = \frac{x^2}{x-1} \), the numerator has a degree of 2, while the denominator has a degree of 1. Since the numerator's degree is greater, there is no horizontal asymptote. Instead, you look for an oblique asymptote. If the degrees were equal, you would simplify the leading coefficients to find the horizontal asymptote, as the value the function approaches as \( x \) approaches infinity.

Oblique asymptotes, sometimes called slant asymptotes, appear when the degree of the numerator is exactly one more than the degree of the denominator in a rational function. Instead of a flat line like a horizontal asymptote, these are diagonal.
To find an oblique asymptote, you perform polynomial long division. For \( y = \frac{x^2}{x-1} \), dividing gives \( y = x + 1 \) with some remainder that becomes insignificant for large values of \( x \). The function will approach the line \( y = x + 1 \) as \( x \) becomes very large in both positive and negative directions.
The oblique asymptote predicts how the function behaves at extreme values of \( x \), acting as a guide to where the tails of the graph will head, providing essential information on the overarching shape of the graph.