Answer
\( y = \frac{-1}{x} \) is a hyperbola located in the second and fourth quadrants with vertical and horizontal asymptotes at \( x = 0 \) and \( y = 0 \) respectively. It is symmetric about the origin and approaches infinity or negative infinity as \( x \) approaches 0 from the right or left, and approaches 0 as \( x \) becomes very large in either direction.
Solution
Certainly! Let's analyze the function provided in part (a):
### (a) \( y = \frac{-1}{x} \)
#### 1. **Function Overview**
The function \( y = \frac{-1}{x} \) is a rational function, specifically a type of hyperbola. It's a reciprocal function multiplied by -1, which affects its graph's orientation.
#### 2. **Domain and Range**
- **Domain**: All real numbers except \( x = 0 \) (since division by zero is undefined).
\[
\text{Domain: } x \in \mathbb{R} \setminus \{0\}
\]
- **Range**: All real numbers except \( y = 0 \).
\[
\text{Range: } y \in \mathbb{R} \setminus \{0\}
\]
#### 3. **Asymptotes**
- **Vertical Asymptote**: \( x = 0 \)
- The function approaches infinity or negative infinity as \( x \) approaches 0 from the right or left.
- **Horizontal Asymptote**: \( y = 0 \) (the x-axis)
- As \( x \) becomes very large (positively or negatively), \( y \) approaches 0.
#### 4. **Graph Characteristics**
- **Quadrants**: The graph of \( y = \frac{-1}{x} \) lies in the second and fourth quadrants because:
- For \( x > 0 \), \( y = -\frac{1}{x} < 0 \) (4th quadrant).
- For \( x < 0 \), \( y = -\frac{1}{x} > 0 \) (2nd quadrant).
- **Symmetry**:
- The graph is **symmetric** with respect to the **origin**. This means if you rotate the graph 180 degrees around the origin, it remains unchanged. Mathematically, this is known as **odd symmetry**, satisfying \( f(-x) = -f(x) \).
#### 5. **Behavior Analysis**
- **As \( x \to 0^+ \)**:
\[
y = \frac{-1}{x} \to -\infty
\]
- **As \( x \to 0^- \)**:
\[
y = \frac{-1}{x} \to +\infty
\]
- **As \( x \to +\infty \)**:
\[
y = \frac{-1}{x} \to 0^-
\]
- **As \( x \to -\infty \)**:
\[
y = \frac{-1}{x} \to 0^+
\]
#### 6. **Plotting Points**
| \( x \) | \( y = \frac{-1}{x} \) |
|---------|------------------------|
| -2 | 0.5 |
| -1 | 1 |
| -0.5 | 2 |
| 0.5 | -2 |
| 1 | -1 |
| 2 | -0.5 |
Plotting these points will help visualize the hyperbola's shape, confirming its placement in the second and fourth quadrants with the described asymptotes.
#### 7. **Summary**
The function \( y = \frac{-1}{x} \) is a hyperbola with:
- **Vertical Asymptote** at \( x = 0 \)
- **Horizontal Asymptote** at \( y = 0 \)
- **Symmetry** about the origin
- Located in the **second and fourth quadrants**
Understanding these properties will aid in graphing the function and analyzing its behavior across different values of \( x \).
Reviewed and approved by the UpStudy tutoring team
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