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Vertical asymptotes are an essential part of understanding rational functions. They occur at points where the function is undefined, primarily when the denominator of a fraction equals zero and the numerator isn't also zero at the same point.

In the case of the function \( r(x) = \frac{x^2 + 2x}{x - 1} \), our denominator is \( x - 1 \). Vertical asymptotes are found by setting the denominator equal to zero which, in this case, is solved by \( x - 1 = 0 \).

Upon solving, we get \( x = 1 \). Thus, a vertical asymptote exists at \( x = 1 \).

This means as \( x \) approaches 1 from either direction, the function \( r(x) \) will approach infinity or negative infinity depending on the side from which you approach. This is crucial behavior to note when sketching the graph of rational functions.

Polynomial long division is a method used to divide one polynomial by another. It's akin to regular long division that's taught in elementary mathematics, but applied to polynomials. This technique is important for finding slant asymptotes in rational functions where the degree of the numerator is greater than the degree of the denominator.

When we performed polynomial long division on \( x^2 + 2x \) divided by \( x - 1 \):

• We divided \( x^2 \) by \( x \) and obtained \( x \).
• We then multiplied \( x \) by \( x - 1 \), which gave us \( x^2 - x \), and subtracted it from \( x^2 + 2x \), leaving \( 3x \).
• Next, \( 3x \) was divided by \( x \) to yield \( 3 \).
• We multiplied \( 3 \) by \( x - 1 \) to get \( 3x - 3 \), and after subtraction, this left us with a remainder of 3.

Thus, \( r(x) \) can be expressed as \( x + 3 + \frac{3}{x-1} \). The outcome of the division is instrumental in determining the function's slant asymptote.

Rational functions are ratios of polynomials, represented as \( f(x) = \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials and \( q(x) eq 0 \).

These functions are characterized by their asymptotic behavior, including vertical and slant asymptotes, and their unique graph shapes. Understanding how to manipulate and sketch them is a vital skill.

In \( r(x) = \frac{x^2 + 2x}{x - 1} \), the numerator \( x^2 + 2x \) and the denominator \( x - 1 \) tell us about the behavior of the function as \( x \) approaches various critical values.

• Vertical asymptotes arise where the denominator zeroes out, unless the numerator is zero at these points too.
• Slant asymptotes occur when the degree of the numerator is precisely one greater than that of the denominator, as is the case here.

Recognizing these aspects helps in graphing and understanding the potential behavior of any rational function.