.

Column A

Column B

A. Zero of the numerator

B. Zero of the denominator

C. Y = 0 if n <d

D. (X, 0)

E. (0, Y)

1. x-intercept

2. y-intercept

3. Zero of the function

4. Vertical asymptote

5. Horizontal asymptote

Let's match the items from **Column A** with the appropriate items from **Column B**.

Here is the reasoning:

1. **X-intercept**: This occurs when \( y = 0 \), which happens when the **numerator** of the function equals zero. This is a point of the form \( (x, 0) \).
   - So, **1 → D** (X, 0).

2. **Y-intercept**: This occurs when \( x = 0 \), meaning the function is evaluated at \( x = 0 \), leading to a point of the form \( (0, y) \).
   - So, **2 → E** (0, Y).

3. **Zero of the function**: This also happens when \( y = 0 \), which again is when the **numerator** is zero.
   - So, **3 → A** Zero of the numerator.

4. **Vertical asymptote**: A vertical asymptote occurs when the **denominator** of a rational function is zero (but the numerator is not zero).
   - So, **4 → B** Zero of the denominator.

5. **Horizontal asymptote**: A horizontal asymptote exists when the degree \( n \) of the numerator is less than the degree \( d \) of the denominator, and the function approaches \( y = 0 \) as \( x \) goes to infinity.
   - So, **5 → C** Y = 0 if n < d.

### Final Matching:
1. **1 → D**
2. **2 → E**
3. **3 → A**
4. **4 → B**
5. **5 → C**

Would you like more details or have any questions about this matching? Here are some related questions to expand your understanding:

1. What happens to the graph of a function at a vertical asymptote?
2. How do you find the x-intercept of a rational function?
3. Can a function have more than one horizontal asymptote?
4. What does it mean for a function to have no x-intercept?
5. How do you determine the behavior of a rational function near its vertical asymptotes?

**Tip:** The degrees of the numerator and denominator in a rational function determine the existence and location of horizontal asymptotes.

Column A

A. Zero of the numerator
B. Zero of the denominator
C. Y = 0 if n < d
D. (X, 0)
E. (0, Y)

Column B

1. x-intercept
2. y-intercept
3. Zero of the function
4. Vertical asymptote
5. Horizontal asymptote

This problem explores key concepts in rational functions, including how to find x-intercepts, y-intercepts, vertical and horizontal asymptotes. Learn how the numerator and denominator affect the graph and behavior of these functions. 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