91Ó°ÊÓ

Vertical asymptotes are locations where the rational function shoots off to positive or negative infinity. They occur when the denominator of the function equals zero, causing the function to be undefined at those points. For the given rational function, \( r(x) = \frac{x^4 - 3x^3 + 6}{x-3} \), we set the denominator, \( x-3 \), equal to zero. Solving \( x-3 = 0 \) gives \( x = 3 \). Hence, there is a vertical asymptote at \( x = 3 \). Remember:

• Vertical asymptotes are vertical lines represented by equations like \( x = a \).
• They tell us where the function's outputs become unbounded.
• They are not influenced by the numerator of the function, but purely by the denominator.

X-intercepts of a rational function are points where the graph crosses the x-axis. This occurs when the function's output is zero. To find these points for the function \( r(x) = \frac{x^4 - 3x^3 + 6}{x-3} \), we set the numerator equal to zero: \( x^4 - 3x^3 + 6 = 0 \). Solving this polynomial without trial solutions involves numerical estimation methods or possibly substitution if a factor can be simplified.

• The x-intercepts occur where the numerator equals zero because \( \frac{0}{b} = 0 \) for any non-zero \( b \).
• These are the points \((x, 0)\) on the graph.
• Graphical methods or numerical solvers may be used for complex polynomials.

Y-intercepts are where the graph of the function crosses the y-axis. At these points, the input \( x = 0 \). For our function \( r(x) \), substitute \( x = 0 \) into the formula:\[ r(0) = \frac{0^4 - 3(0)^3 + 6}{0-3} = -2 \]Thus, the y-intercept is at the coordinate \((0, -2)\). It's important to remember:

• Y-intercepts are always found by evaluating the function at \( x = 0 \).
• They provide insight into the initial value of the function.
• An interception occurs at a single vertical spot on the y-axis.

Rational functions may exhibit local maxima or minima, which are the peaks or valleys on their graphs. To find these extrema, we use the derivative of the function to identify critical points. For the function \( r(x) \), applying the quotient rule gives us:\[ r'(x) = \frac{(4x^3 - 9x^2)(x - 3) - (x^4 - 3x^3 + 6)}{(x-3)^2} \]To find potential extrema, solve \( r'(x) = 0 \) and analyze these critical points:

• Local maxima/minima occur where the derivative changes signs.
• Use a sign chart or second derivative test to determine the nature of each critical point.
• Critical points need further analysis to confirm they are true local maxima or minima.

Polynomial long division is a technique used to divide a polynomial by another polynomial of lower degree. This process can help simplify a rational function or understand its end behavior. For \( r(x) = \frac{x^4 - 3x^3 + 6}{x-3} \), we perform long division as follows:1. Divide the highest degree term of the numerator \( x^4 \) by \( x \), the leading term of the denominator.2. Multiply the entire denominator by the result and subtract from the original polynomial.3. Repeat the process with the new polynomial formed by the subtraction until no remainder degree exceeds that of the denominator.In this particular case, the result of the division is \( x^3 \), confirming that the higher degree terms \( x^3 \) dictate the end behavior of the function \( r(x) \).

• Long division helps isolate and simplify polynomial expressions.
• The quotient gained from long division corresponds to the function's behavior at infinity.
• This method is analogous to dividing numbers, focusing on highest degree terms first.