https://file.aiquickdraw.com/mathbot/file/1729471758461lk97s43q.jpg

To determine the slant asymptote of the function \( f(x) = \frac{4x^2 - x - 2}{x - 1} \), we use polynomial division because a slant (oblique) asymptote occurs when the degree of the numerator is exactly one higher than the degree of the denominator, as it is here.

### Steps:

1. **Perform Polynomial Long Division**:
   Divide \( 4x^2 - x - 2 \) by \( x - 1 \).

   - First, divide \( 4x^2 \) by \( x \), which gives \( 4x \).
   - Multiply \( 4x \) by \( x - 1 \), which gives \( 4x^2 - 4x \).
   - Subtract \( 4x^2 - 4x \) from \( 4x^2 - x - 2 \), yielding \( 3x - 2 \).

   - Next, divide \( 3x \) by \( x \), which gives \( +3 \).
   - Multiply \( 3 \) by \( x - 1 \), which gives \( 3x - 3 \).
   - Subtract \( 3x - 3 \) from \( 3x - 2 \), yielding a remainder of \( +1 \).

2. **Obtain the Equation of the Slant Asymptote**:
   The result of the division is \( f(x) = 4x + 3 + \frac{1}{x - 1} \).

   As \( x \to \infty \), the remainder term \( \frac{1}{x - 1} \to 0 \), so the slant asymptote is given by:

   \[
   y = 4x + 3
   \]

Thus, the slant asymptote is \( y = 4x + 3 \).

Would you like further details or have any questions?

### Related Questions:
1. How can polynomial division be used for other rational functions?
2. What is the difference between vertical and slant asymptotes?
3. Can a rational function have both a horizontal and a slant asymptote?
4. What happens to the graph of \( f(x) \) near the vertical asymptote at \( x = 1 \)?
5. How does the remainder term \( \frac{1}{x - 1} \) affect the behavior of \( f(x) \)?

### Tip:
Always check the degree of the numerator and denominator when looking for slant asymptotes—this will guide you to the right method!

Determine the slant asymptote of the graph of f(x) = (4x^2 - x - 2) / (x - 1).

Learn how to determine the slant asymptote for the rational function f(x) = (4x^2 - x - 2) / (x - 1) using polynomial long division. This solution provides a step-by-step guide and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation