https://file.aiquickdraw.com/mathbot/file/17267946270231xrjw5ml.jpg

The function given is:

\[
y = \frac{x^2 + 4}{x^2 - 1}
\]

### Vertical Asymptotes:
Vertical asymptotes occur where the denominator is zero (i.e., where the function is undefined), provided the numerator is not zero at the same points.

The denominator is \( x^2 - 1 \), which can be factored as:
\[
x^2 - 1 = (x - 1)(x + 1)
\]
Setting the denominator equal to zero gives the vertical asymptotes:
\[
x = 1 \quad \text{and} \quad x = -1
\]

### Horizontal Asymptotes:
To find the horizontal asymptote, we analyze the degree of the numerator and denominator. Both the numerator and denominator are degree 2 polynomials.

For rational functions where the degree of the numerator and denominator are equal, the horizontal asymptote is found by taking the ratio of the leading coefficients. The leading coefficient of \(x^2\) in both the numerator and denominator is 1, so the horizontal asymptote is:
\[
y = \frac{1}{1} = 1
\]

### Conclusion:
- Vertical asymptotes: \(x = -1\) and \(x = 1\)
- Horizontal asymptote: \(y = 1\)

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. How would the asymptotes change if the numerator were \(x^3 + 4\)?
2. What happens to the graph near a vertical asymptote?
3. Can a rational function have no vertical asymptotes? If so, when?
4. How can you determine a slant asymptote for a rational function?
5. How do holes in the graph of a rational function relate to vertical asymptotes?

**Tip:** To identify vertical asymptotes, always check where the denominator equals zero, as these points indicate where the function is undefined.

Find the horizontal and vertical asymptotes of the curve given below. List them in increasing order. If there is no such asymptote, enter 'N'. y = (x^2 + 4) / (x^2 - 1)

Learn how to find the vertical and horizontal asymptotes for the rational function y = (x^2 + 4)/(x^2 - 1). Understand how to analyze rational functions, identify asymptotes, and apply relevant algebraic techniques. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation