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A slant asymptote, also known as an oblique asymptote, occurs in a rational function when the degree of the numerator is one more than that of the denominator. This means the graph of the function will approach a line that is neither horizontal nor vertical as x heads towards positive or negative infinity.

To find the slant asymptote for a given function, we perform polynomial long division on the function. For example, in the rational function \(r(x) = \frac{x^3+4}{2x^2+x-1}\), the numerator has a degree of 3, while the denominator has a degree of 2. By dividing \(x^3 + 4\) by \(2x^2 + x - 1\), we discovered that the slant asymptote is \(y = \frac{1}{2}x\). This line describes the end behavior of the function, indicating a linear trajectory as the function values grow large or small.

Vertical asymptotes occur in a rational function wherever the denominator is zero and the numerator is non-zero at those points. They indicate that the function approaches positive or negative infinity as it gets closer to a certain x-value.

To identify vertical asymptotes in the function \(r(x) = \frac{x^3+4}{2x^2+x-1}\), we need to solve for when \(2x^2 + x - 1 = 0\). Using the quadratic formula, we find the solutions to be \(x = \frac{1}{2}\) and \(x = -1\). At these points, the function becomes undefined, and its value tends toward infinity.

• For rational functions, each vertical asymptote corresponds to a root of the denominator that does not cancel with the numerator.
• The curve on the graph will tend to rise or fall sharply as it approaches these x-values.

Polynomial long division is a technique used to divide polynomial expressions, similar to long division with numbers. It helps to find slant asymptotes or simplify expressions.

Let's use our example: divide \(x^3 + 4\) by \(2x^2 + x - 1\). Think of this as a way to "fit" the smaller polynomial into the larger one. We repeatedly divide the leading terms, subtract, and bring down the next terms, much like traditional long division.

The result for our function is \(\frac{1}{2}x\) with a remainder, suggesting that the non-remainder part forms the slant asymptote's equation: \(y = \frac{1}{2}x\).

• This quotient (\(\frac{1}{2}x\)) approximates the behavior of the function graph as \(x\) heads to infinity.
• The remainder in the division affects the approximation's accuracy but typically diminishes with larger x-values.

Graphing a rational function involves combining several aspects: the location of vertical and slant (or horizontal) asymptotes, x-intercepts, y-intercepts, and overall behavior.

For instance, with \(r(x) = \frac{x^3+4}{2x^2+x-1}\), we already discovered its slant asymptote \(y = \frac{1}{2}x\) and vertical asymptotes at \(x = \frac{1}{2}\) and \(x = -1\). These guides indicate the boundaries and directional tendencies of the function.

• As \(x\) approaches each vertical asymptote, the graph should diverge to \(+\infty\) or \(-\infty\).
• As \(x\) approaches \(+\infty\) or \(-\infty\), the function approaches its slant asymptote.
• Sketching the graph helps visualize these asymptotic behaviors.
• Plot additional points near key features to verify the curve's shape and direction.