what is a function with a horizontal asymptote at y=3, vertical asymptotes at x=0 and x=2

A function with a **horizontal asymptote at \( y = 3 \)** and **vertical asymptotes at \( x = 0 \)** and \( x = 2 \) could be a **rational function** of the form:

\[
f(x) = \frac{3(x - a)}{(x)(x - 2)}
\]

### Explanation:

1. **Horizontal Asymptote** at \( y = 3 \):  
   - The horizontal asymptote occurs when the degree of the numerator is equal to the degree of the denominator. The ratio of the leading coefficients of the numerator and denominator determines the horizontal asymptote. In this case, the horizontal asymptote is \( y = 3 \), so the leading coefficient of the numerator must be 3.
   
2. **Vertical Asymptotes** at \( x = 0 \) and \( x = 2 \):  
   - Vertical asymptotes occur where the denominator is zero, so we need \( x = 0 \) and \( x = 2 \) to be the roots of the denominator. Thus, the denominator must include \( (x)(x - 2) \).

### Example function:
A simple function that satisfies these conditions is:

\[
f(x) = \frac{3}{x(x - 2)}
\]

This function has:
- A horizontal asymptote at \( y = 3 \) as \( x \to \infty \).
- Vertical asymptotes at \( x = 0 \) and \( x = 2 \).

Would you like further details or have any questions?

Here are 5 related questions:
1. What are the steps to determine horizontal and vertical asymptotes for rational functions?
2. How do you find the oblique (slant) asymptotes of a function?
3. Can you describe how the behavior of a function changes near its vertical asymptotes?
4. How can we adjust this function to shift the horizontal asymptote to a different value, like \( y = 5 \)?
5. What happens to the graph of this function as \( x \to -\infty \)?

**Tip:** A horizontal asymptote shows the end behavior of a function as \( x \to \infty \) or \( x \to -\infty \).

What is a function with a horizontal asymptote at y=3, vertical asymptotes at x=0 and x=2?

Explore how to find a rational function with a horizontal asymptote at y=3 and vertical asymptotes at x=0 and x=2. This guide explains key algebraic concepts and formulas, 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