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 \( x \).

Hogar Banco de preguntas Matemáticas Álgebra

Pregunta

(a)

Ask by Reese Gardner. in Malaysia

Jan 20,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

is a hyperbola located in the second and fourth quadrants with vertical and horizontal asymptotes at

and

respectively. It is symmetric about the origin and approaches infinity or negative infinity as

approaches 0 from the right or left, and approaches 0 as

becomes very large in either direction.

Solución

Certainly! Let’s analyze the function provided in part (a):

(a)

1. Function Overview

The function

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 (since division by zero is undefined).
Range: All real numbers except .

3. Asymptotes

Vertical Asymptote:
- The function approaches infinity or negative infinity as approaches 0 from the right or left.
Horizontal Asymptote: (the x-axis)
- As becomes very large (positively or negatively), approaches 0.

4. Graph Characteristics

Quadrants: The graph of lies in the second and fourth quadrants because:
- For , (4th quadrant).
- For , (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 .

5. Behavior Analysis

As :
As :
As :
As :

6. Plotting Points


-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

is a hyperbola with:

Vertical Asymptote at
Horizontal Asymptote at
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

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Bonus Knowledge

Did you know that the equation

describes a hyperbola? This fascinating shape comes from a family of conic sections and has interesting properties, including asymptotes, which are the lines that the curve approaches but never touches. The asymptotes for this equation are the x-axis and y-axis, creating a visual intersection point in the Cartesian plane!

When grappling with this equation, a common mistake is forgetting the domain restrictions. Since you can’t divide by zero, x cannot equal zero. This oversight might lead to incorrect graphing, resulting in a broken line in your visualization! Always check for such restrictions to ensure your graphs are accurate and complete.

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