Let's analyze the rational function \( f(x) = \frac{x+3}{x-4} \):
1. **Holes:**
- There is a hole in the graph if there is a factor common to both the numerator and the denominator that cancels out. In this case, there are no common factors, so there are no holes.
2. **Vertical Asymptotes:**
- Vertical asymptotes occur where the denominator equals zero. In this case, \( x - 4 = 0 \) gives \( x = 4 \). Therefore, there is a vertical asymptote at \( x = 4 \).
3. **Horizontal Asymptotes:**
- For rational functions, the degree of the numerator and denominator determines horizontal asymptotes. In this case, both the numerator and denominator have the same degree (1), so there is a horizontal asymptote. To find it, compare the leading terms: \( \frac{x}{x} \). Therefore, there is a horizontal asymptote at \( y = 1 \).
4. **X-Intercepts:**
- X-intercepts occur when the numerator is equal to zero. In this case, \( x + 3 = 0 \) gives \( x = -3 \). Therefore, there is an x-intercept at \( x = -3 \).
5. **Y-Intercepts:**
- Y-intercepts occur when \( x = 0 \). Plug in \( x = 0 \) into the function: \( f(0) = \frac{3}{-4} = -\frac{3}{4} \). Therefore, there is a y-intercept at \( (0, -\frac{3}{4}) \).
In summary:
- Holes: None
- Vertical Asymptotes: \( x = 4 \)
- Horizontal Asymptotes: \( y = 1 \)
- X-Intercepts: \( x = -3 \)
- Y-Intercepts: \( (0, -\frac{3}{4}) \)
