https://file.aiquickdraw.com/mathbot/file/17300825149627sfv95rr.jpg

To find and graph all asymptotes of the rational function 

\[
f(x) = \frac{6x^2 + 7x - 9}{2x + 1},
\]

we need to determine both vertical and horizontal (or slant) asymptotes.

### 1. Vertical Asymptote
Vertical asymptotes occur where the denominator is zero, as long as these points do not cancel with terms in the numerator.

The denominator is \(2x + 1\). Setting it equal to zero:
\[
2x + 1 = 0 \Rightarrow x = -\frac{1}{2}.
\]
Therefore, there is a vertical asymptote at \(x = -\frac{1}{2}\).

### 2. Slant Asymptote (Oblique Asymptote)
Since the degree of the numerator (2) is exactly one degree higher than the degree of the denominator (1), there will be a slant asymptote. We find it by performing polynomial long division on \(6x^2 + 7x - 9\) divided by \(2x + 1\).

**Step-by-step division:**

1. Divide the leading term \(6x^2\) by \(2x\), which gives \(3x\).
2. Multiply \(3x\) by \(2x + 1\), resulting in \(6x^2 + 3x\).
3. Subtract \(6x^2 + 3x\) from \(6x^2 + 7x - 9\) to get \(4x - 9\).
4. Divide \(4x\) by \(2x\) to get \(+2\).
5. Multiply \(2\) by \(2x + 1\), resulting in \(4x + 2\).
6. Subtract \(4x + 2\) from \(4x - 9\) to get \(-11\).

Thus, the division gives:
\[
f(x) = 3x + 2 + \frac{-11}{2x + 1}.
\]

As \(x \to \infty\) or \(x \to -\infty\), the term \(\frac{-11}{2x + 1} \to 0\), so the slant asymptote is:
\[
y = 3x + 2.
\]

### Summary
- **Vertical Asymptote**: \(x = -\frac{1}{2}\).
- **Slant Asymptote**: \(y = 3x + 2\).

Would you like a detailed graph of these asymptotes, or have any questions?

---

#### Related Questions
1. How do you find the vertical asymptotes for rational functions with more complex denominators?
2. What is the difference between horizontal and slant asymptotes?
3. Can a rational function have both a horizontal and a slant asymptote?
4. What are the steps to perform polynomial long division?
5. How do asymptotes affect the behavior of the graph of a rational function?

#### Tip
When the degree of the numerator is exactly one more than the degree of the denominator, always look for a slant asymptote by performing polynomial division.

Graph all asymptotes of the rational function f(x) = (6x^2 + 7x - 9) / (2x + 1).

Learn how to graph the asymptotes of the rational function f(x) = (6x^2 + 7x - 9) / (2x + 1). This solution provides a step-by-step approach to finding vertical and slant asymptotes using polynomial division, suitable for high school 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