To find the asymptotes of the rational function \( Y = \frac{(x - 1)(x - 3)}{x^2} \), we will analyze both the vertical and horizontal asymptotes.

### 1. Vertical Asymptotes:
Vertical asymptotes occur where the denominator of the function equals zero, provided the numerator does not also equal zero at those points.

- The denominator is \( x^2 \), which is zero when \( x = 0 \).

Thus, there is a vertical asymptote at:
\[ x = 0. \]

### 2. Horizontal Asymptotes:
Horizontal asymptotes describe the behavior of the function as \( x \) approaches infinity or negative infinity. For rational functions, the horizontal asymptote is determined by the degree of the polynomials in the numerator and denominator.

- The degree of the numerator, \( (x - 1)(x - 3) = x^2 - 4x + 3 \), is 2.
- The degree of the denominator, \( x^2 \), is also 2.

When the degrees of the numerator and denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients of the numerator and denominator.

Here, the leading coefficient of the numerator is 1 (from \( x^2 \)) and the leading coefficient of the denominator is 1 (from \( x^2 \)).

Thus, the horizontal asymptote is:
\[ y = \frac{1}{1} = 1. \]

### Summary of Asymptotes:
- **Vertical asymptote:** \( x = 0 \)
- **Horizontal asymptote:** \( y = 1 \)

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. How do you find slant asymptotes for rational functions?
2. What happens if the numerator has a higher degree than the denominator?
3. How does a hole in the graph differ from a vertical asymptote?
4. How do vertical asymptotes affect the limits of the function?
5. Can you have more than one horizontal asymptote?

**Tip:** When identifying horizontal asymptotes, always compare the degrees of the numerator and denominator!

Explore how to find the vertical and horizontal asymptotes of the rational function Y = (x-1)(x-3)/x^2. This detailed solution covers key concepts in algebra, including vertical and horizontal asymptotes, with step-by-step explanations. 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