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To solve this problem, we are tasked with finding the asymptotes of three given hyperbolas, each represented by a rational function.

The general form of a rational function like these is:
\[
y = \frac{A}{x} + B
\]
where \( A \) and \( B \) are constants.

### Key Idea:
For hyperbolas represented by rational functions, the vertical asymptote is where the denominator is zero, which is \( x = 0 \). The horizontal asymptote is determined by the behavior of the function as \( x \) approaches infinity, which will be \( y = B \), where \( B \) is the constant term added to the rational part of the function.

### Now, let's go case by case.

#### (i) \( y = \frac{1}{x} + 5 \)

- **Vertical Asymptote:** The vertical asymptote occurs when the denominator is zero, i.e., \( x = 0 \).
- **Horizontal Asymptote:** As \( x \to \infty \), the term \( \frac{1}{x} \) approaches zero, so the horizontal asymptote is \( y = 5 \).

Thus, the asymptotes are:
- Vertical: \( x = 0 \)
- Horizontal: \( y = 5 \)

#### (ii) \( y = \frac{3}{x} - 4 \)

- **Vertical Asymptote:** The denominator is zero at \( x = 0 \), so the vertical asymptote is \( x = 0 \).
- **Horizontal Asymptote:** As \( x \to \infty \), \( \frac{3}{x} \) approaches zero, so the horizontal asymptote is \( y = -4 \).

Thus, the asymptotes are:
- Vertical: \( x = 0 \)
- Horizontal: \( y = -4 \)

#### (iii) \( y = \frac{-2}{x} + 1 \)

- **Vertical Asymptote:** Again, \( x = 0 \) is the vertical asymptote because the denominator is zero there.
- **Horizontal Asymptote:** As \( x \to \infty \), \( \frac{-2}{x} \) approaches zero, leaving the horizontal asymptote at \( y = 1 \).

Thus, the asymptotes are:
- Vertical: \( x = 0 \)
- Horizontal: \( y = 1 \)

### Summary of Asymptotes:

1. \( y = \frac{1}{x} + 5 \): 
   - Vertical Asymptote: \( x = 0 \)
   - Horizontal Asymptote: \( y = 5 \)
   
2. \( y = \frac{3}{x} - 4 \): 
   - Vertical Asymptote: \( x = 0 \)
   - Horizontal Asymptote: \( y = -4 \)
   
3. \( y = \frac{-2}{x} + 1 \): 
   - Vertical Asymptote: \( x = 0 \)
   - Horizontal Asymptote: \( y = 1 \)

Would you like more details or have any other questions?

### Related Questions:
1. What are the general steps for finding vertical and horizontal asymptotes for any rational function?
2. How do the constants in the rational function affect the location of the asymptotes?
3. What is the difference between slant asymptotes and horizontal asymptotes?
4. How can we find the asymptotes of more complicated rational functions, such as those with quadratic terms in the denominator?
5. How do we sketch a rational function once the asymptotes are known?

### Tip:
When finding horizontal asymptotes for rational functions, always check the degrees of the numerator and the denominator—this determines if the horizontal asymptote exists and where it is located.

Without graphing, write the equations of the asymptotes of these hyperbolas: i. y = 1/x + 5, ii. y = 3/x - 4, iii. y = -2/x + 1

This guide covers how to determine the vertical and horizontal asymptotes of the hyperbolas represented by rational functions like y = 1/x + 5, y = 3/x - 4, and y = -2/x + 1. Learn the step-by-step method of identifying asymptotes without graphing, suitable for Grades 10-12.

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