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A slant asymptote, also known as an oblique asymptote, occurs when the degree of the numerator in a rational function is exactly one higher than the degree of the denominator. This results in the graph having a diagonal line as it extends towards positive or negative infinity. In this exercise, the function is given as \[ f(x) = \frac{x^2}{x-3} \]Here, the numerator, \(x^2\), is a polynomial of degree 2, while the denominator, \(x-3\), is of degree 1. Since the degree of the numerator is one more than the degree of the denominator, we can expect a slant asymptote. To find this asymptote, polynomial long division is used, which results in \(y = x + 3\) as the slant asymptote. This diagonal line, \(y = x + 3\), shows the general direction the graph follows as \(x\) goes towards \(\pm \infty\).

Polynomial long division works similarly to the long division of numbers. It is used to divide two polynomials and simplifies the operation to determine the slant asymptote when the degrees of polynomials are different. For the function \(f(x) = \frac{x^2}{x-3} \), we perform polynomial long division to obtain 1. Divide the leading term of the numerator (\(x^2\)) by the leading term of the denominator (\(x\)), which gives you \(x\).2. Multiply the entire divisor \(x - 3\) by this quotient \(x\) to get \(x^2 - 3x\). 3. Subtract \(x^2 - 3x\) from \(x^2\) to get \(3x\). 4. Repeat the process with \(3x\), dividing by \(x\), you obtain \(+3\). 5. Multiply \(x - 3\) by \(+3\) to obtain \(3x - 9\), and subtract from \(3x\) to get a remainder of \(9\).6. Finally, since the remainder \(9/(x-3)\) decreases to zero as \(x\) increases, it confirms \(y = x + 3\) as the slant asymptote.This method not only helps in identifying asymptotes but also in expressing the original rational function in simpler terms.

To sketch the graph of a function with asymptotes, it's important to plot these key features. 1. **Plot the Slant Asymptote**: For the function \(f(x) = \frac{x^2}{x-3}\), the slant asymptote is \(y = x + 3\). Draw this as a guideline of where the graph will trend towards.2. **Identify Other Asymptotes**: The function has a vertical asymptote at \(x = 3\) where the denominator becomes zero. The graph will approach but never touch this line.3. **Sketch the Behavior**: - For \(x > 3\), the function \(f(x)\) will approach the slant asymptote from above, suggesting that the graph goes upwards. - For \(x < 3\), the function \(f(x)\) approaches the slant asymptote from below. The graph trends downwards in this section.4. **Plot Points**: Additional points on both sides of \(x=3\) can be plotted to better understand how the graph behaves between and around asymptotes.Graph sketching combines the understanding of asymptotic behaviors with specific points to provide a more comprehensive picture of the function's nature along its domain.