Questions: Identify the horizontal or slant asymptote for f(x)=(6x^2-x)/(x-2). A. y=6x+13 B. y=6 C. y=6x+11 D. y=0

Identify the horizontal or slant asymptote for f(x)=(6x^2-x)/(x-2). A. y=6x+13 B. y=6 C. y=6x+11 D. y=0
Transcript text: Identify the horizontal or slant asymptote for $f(x)=\frac{6 x^{2}-x}{x-2}$. A. $y=6 x+13$ B. $y=6$ C. $y=6 x+11$ D. $y=0$
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Solution

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Solution Steps

Step 1: Determine the Degree of the Polynomial

The function given is \( f(x) = \frac{6x^2 - x}{x - 2} \). To find the horizontal or slant asymptote, we first need to determine the degree of the numerator and the denominator.

  • The degree of the numerator \( 6x^2 - x \) is 2.
  • The degree of the denominator \( x - 2 \) is 1.
Step 2: Identify the Type of Asymptote

Since the degree of the numerator (2) is greater than the degree of the denominator (1), the function has a slant (oblique) asymptote, not a horizontal asymptote.

Step 3: Find the Slant Asymptote

To find the slant asymptote, perform polynomial long division of the numerator by the denominator:

  1. Divide the leading term of the numerator \( 6x^2 \) by the leading term of the denominator \( x \), which gives \( 6x \).
  2. Multiply \( 6x \) by \( x - 2 \) to get \( 6x^2 - 12x \).
  3. Subtract \( 6x^2 - 12x \) from \( 6x^2 - x \) to get \( 11x \).
  4. Divide \( 11x \) by \( x \) to get \( 11 \).
  5. Multiply \( 11 \) by \( x - 2 \) to get \( 11x - 22 \).
  6. Subtract \( 11x - 22 \) from \( 11x \) to get \( 22 \).

The quotient is \( 6x + 11 \), which is the equation of the slant asymptote.

Final Answer

The slant asymptote of the function \( f(x) = \frac{6x^2 - x}{x - 2} \) is \( y = 6x + 11 \).

\[ \boxed{\text{C. } y = 6x + 11} \]

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