Question

Graph the function y = -2/x

Answer 1

To graph the function \( y = -\frac{2}{x} \), we can follow these steps:

1. **Identify the Asymptotes**: The function has a vertical asymptote at \( x = 0 \) (since the function is undefined when \( x = 0 \)) and a horizontal asymptote at \( y = 0 \) (as \( x \) approaches infinity or negative infinity).

2. **Plot Points**: You can calculate some key points to help in sketching the graph:
- When \( x = 1 \), \( y = -\frac{2}{1} = -2 \).
- When \( x = -1 \), \( y = -\frac{2}{-1} = 2 \).
- When \( x = 2 \), \( y = -\frac{2}{2} = -1 \).
- When \( x = -2 \), \( y = -\frac{2}{-2} = 1 \).
- As \( x \) approaches 0 from the positive side (e.g., \( 0.1 \)), \( y \) approaches negative infinity.
- As \( x \) approaches 0 from the negative side (e.g., \( -0.1 \)), \( y \) approaches positive infinity.

3. **Shape of the Graph**: The function is hyperbolic, and since the coefficient of \( x \) is negative, both branches of the hyperbola will lie in the second quadrant (where \( x < 0 \) and \( y > 0 \)) and the fourth quadrant (where \( x > 0 \) and \( y < 0 \)).

4. **Draw the Axes**: Set up your coordinate plane with the horizontal axis (x-axis) and vertical axis (y-axis). Mark the asymptotes with dashed lines.

5. **Sketch the Graph**: Using the points and asymptotes, draw the curve:
- The curve approaches the vertical asymptote (y-axis) but never touches it.
- The curve also approaches the horizontal asymptote (x-axis) as \( x \) moves to positive or negative infinity.

In summary, the graph of \( y = -\frac{2}{x} \) consists of two branches:
- One in the second quadrant that moves up and to the left.
- The other in the fourth quadrant that moves down and to the right.

Here’s a simple representation:

```
|
2 | x
| /
| /
------|-------*-------
| /
-2 | /
| /
| /
-1 | /
| /
|/
-2 |
------
```

Where the asterisk (*) is at the point (1, -2) and (−1, 2), illustrating the hyperbolic nature of the graph.