find the zeros, vertical asymptotes, holes, and y intercepts of 1/x+4 -3

To analyze the function \( f(x) = \frac{1}{x + 4} - 3 \), we will find the following:

### 1. **Zeros** (where \( f(x) = 0 \)):
Set \( f(x) = 0 \) to find the zeros.
\[
\frac{1}{x + 4} - 3 = 0
\]
Add 3 to both sides:
\[
\frac{1}{x + 4} = 3
\]
Multiply both sides by \( x + 4 \):
\[
1 = 3(x + 4)
\]
Solve for \( x \):
\[
1 = 3x + 12
\]
\[
3x = 1 - 12
\]
\[
3x = -11
\]
\[
x = -\frac{11}{3}
\]
Thus, the zero is at \( x = -\frac{11}{3} \).

### 2. **Vertical Asymptote**:
The vertical asymptote occurs where the denominator is zero. The denominator of \( \frac{1}{x + 4} \) is zero when:
\[
x + 4 = 0
\]
\[
x = -4
\]
So, the vertical asymptote is at \( x = -4 \).

### 3. **Holes**:
Holes occur where there are common factors between the numerator and denominator, which cancel out. The function \( f(x) = \frac{1}{x+4} - 3 \) has no common factors to cancel, so **there are no holes** in this function.

### 4. **Y-Intercept** (where \( x = 0 \)):
To find the y-intercept, set \( x = 0 \):
\[
f(0) = \frac{1}{0 + 4} - 3 = \frac{1}{4} - 3 = \frac{1}{4} - \frac{12}{4} = \frac{-11}{4}
\]
Thus, the y-intercept is \( \left( 0, \frac{-11}{4} \right) \).

### Summary:
- **Zero**: \( x = -\frac{11}{3} \)
- **Vertical asymptote**: \( x = -4 \)
- **Holes**: None
- **Y-intercept**: \( \left( 0, \frac{-11}{4} \right) \)

Would you like any further details or clarification?

Here are 5 related questions you might want to explore:
1. How do you graph a rational function like this one?
2. What is the behavior of \( f(x) \) near the vertical asymptote?
3. How do you determine horizontal asymptotes for rational functions?
4. Can rational functions have slant asymptotes?
5. How do transformations affect rational functions like \( \frac{1}{x+4} - 3 \)?

**Tip**: When analyzing rational functions, always check for holes by factoring both the numerator and denominator, as holes occur when terms cancel out.

Find the zeros, vertical asymptotes, holes, and y intercepts of 1/(x+4) - 3.

Explore how to find the zeros, vertical asymptotes, and y-intercepts of the rational function f(x) = 1/(x+4) - 3. This problem covers key concepts such as determining zeros, identifying vertical asymptotes, and finding the y-intercept. Suitable for high school students in Grades 9-12.

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