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Vertical asymptotes in a rational function occur where the function’s denominator equals zero and is not canceled by the numerator. They represent the values of \( x \) where the function approaches infinity. In our exercise, we have the rational function \[y = \frac{x^4 - 3x^3 + x^2 - 3x + 3}{x^2 - 3x}\]We want to find where the denominator, \( x^2 - 3x \), equals zero. First, factor this expression to get \( x(x-3) = 0 \). Solving this equation gives us the solutions \( x = 0 \) and \( x = 3 \). These are the locations of our vertical asymptotes since neither of these values zero out the numerator. Remember, vertical asymptotes do not cross the graph. They signify boundaries the graph approaches but never touches.

Polynomial long division helps in simplifying complex rational expressions and understanding their behavior at extremes. Here, we need to divide the numerator \( x^4 - 3x^3 + x^2 - 3x + 3 \) by the denominator \( x^2 - 3x \). This type of division involves repeatedly dividing the terms of the numerator by the terms in the denominator, much like arithmetic long division.First, divide the highest degree term in the numerator, \( x^4 \), by the highest degree term in the divisor, \( x^2 \). This gives us the first term in the quotient, \( x^2 \). Multiply \( x^2 \) by \( x^2 - 3x \) and subtract the result from the original numerator. Repeat this process with the new remainder.The result of this division tells us that the polynomial \( y = x^2 \) mirrors the end behavior of the original function. This means that, for very large values of \( x \), the rational function behaves similarly to the simpler polynomial function.

Finding the \( x \)-intercepts of a rational function involves setting the numerator equal to zero and solving for \( x \). For our function:\[x^4 - 3x^3 + x^2 - 3x + 3 = 0\]Solving this quartic equation can be tricky. In these cases, using a graphing calculator or computer algebra system (CAS) can be practical to find approximate solutions. These tools graph the function and identify where the graph crosses the \( x \)-axis.Remember, the \( x \)-intercepts are the roots of the numerator, signifying the points on the graph where the function equals zero. These points may not always be integer or readily apparent, which is why numerical solutions or approximations can be helpful.

The end behavior of a function describes how it behaves as \( x \to \pm \infty \). For rational functions, this can generally be understood through the degree of the numerator and denominator. If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. However, in our case, polynomial long division already simplified this to \( y = x^2 \).This polynomial indicates that as \( x \to \infty \) or \( x \to -\infty \), the graph of the rational function approaches the behavior of \( y = x^2 \). Both the rational function and this polynomial will grow larger indefinitely, like a parabola opening upwards.Graphically, when both functions are plotted, they should overlay each other at extreme domains, confirming that they share the same end behavior. This can be verified by graphing both the rational and the resulting polynomial within a sufficiently large window, illustrating their overlapping trajectories as \( x \) moves away from zero.