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Vertical asymptotes in rational functions denote the values of the variable where the function's value becomes infinitely large or small, giving a sense of where the graph of the function approaches but never touches.

For instance, in the given exercise where we have the function \( f(x)=\frac{x-1}{x-2} \), \( x=2 \) is the vertical asymptote. At this point, the denominator of the function equals zero, making the function undefined and the graph of the function tends toward infinity (or negative infinity) as \( x \) approaches the vertical asymptote from either side of the graph.

Remember, a function may have multiple vertical asymptotes, as seen in some of the given exercises where the denominators contain more than one factor that could be set to zero. Vertical asymptotes are crucial for sketching the graph of rational functions.

Horizontal asymptotes, on the other hand, refer to the line that the graph of a rational function approaches as the variable either goes to positive or negative infinity. It's a way to determine the end behavior of a function.

In the exercise solutions, the horizontal asymptotes are all represented by the equation \( y= \) some constant. This asymptote tells us the value that \( f(x) \) gets closer to, but doesn't actually reach, as \( x \) increases or decreases without bound. For instance, the function \( f(x)=\frac{x-2}{x+1} \) has a horizontal asymptote at \( y=0 \), indicating that as \( x \) becomes very large or very negative, the value of \( f(x) \) approaches zero.

The horizontal asymptotes depend largely on the degrees of the polynomials in the numerator and denominator of the function. If the degree of the numerator is equal to or less than the degree of the denominator, the horizontal asymptote will be at \( y=0 \); if the degrees are equal, the horizontal asymptote will be at \( y= \) the ratio of the leading coefficients.

The zeros of a function are the values of the variable that make the function equal to zero. These are the points where the graph of the function will cross the horizontal axis.

For a rational function, zeros are found by setting the numerator equal to zero and solving for the variable. For example, in our function \( f(x)=\frac{x-2}{x+1} \), the numerator \( x-2 = 0 \) gives us the zero at \( x=2 \). Each zero translates into an \( x \)-intercept on the graph of the function.

Zeros are important for understanding the behavior and characteristics of the function, as well as for graphing. They provide key points through which the graph will pass, offering a framework for the rest of the graph. Typically, the behavior of rational functions near their zeros will depend on the order of the zero — if the zero has an even multiplicity, the graph will touch the axis and turn back; if the zero is of odd multiplicity, the graph will cross the axis.

Graphing rational functions involves a step-by-step process that takes into account the zeros, vertical and horizontal asymptotes, and often other features such as y-intercepts or symmetry.

The rational functions provided in the exercise illustrate how knowing the zeros and asymptotes can form an initial 'skeleton' of the graph. To accurately sketch these functions, you start by plotting the asymptotes as dashed lines that the graph will never touch. Next, plot the zeros where the function intersects the \( x \)-axis.

After placing these foundational elements on the graph, it is possible to sketch the general shape of the function. It's important to note how the function behaves near the asymptotes and zeros: for example, whether it approaches the asymptotes from above or below, and how it passes through the zeros. By understanding these elements, graphing any rational function can become a manageable, step-by-step process.