show how a -INF and +INF graph looks like and explain why it is

To illustrate how the graph of \( -\infty \) (negative infinity) and \( +\infty \) (positive infinity) looks like, I'll explain the concepts first, then describe the graph, and follow that with an explanation.

### **Understanding \( -\infty \) and \( +\infty \)**
1. **Negative Infinity (\( -\infty \))**: This occurs when the values on the graph approach infinitely large negative numbers as the variable (typically \( x \) or \( y \)) moves in a certain direction.
   
   - Example: As \( x \to -\infty \), it means that \( x \) is decreasing without bound (i.e., getting more and more negative).

2. **Positive Infinity (\( +\infty \))**: This happens when the values on the graph approach infinitely large positive numbers as the variable moves in a specific direction.

   - Example: As \( x \to +\infty \), it means \( x \) is increasing without bound (i.e., getting more and more positive).

### **Graphical Representation**
On a Cartesian plane:

- When \( x \to -\infty \), the graph stretches far to the left.
- When \( x \to +\infty \), the graph stretches far to the right.

Now, we also need to discuss how the **y-values** (the function's values) behave:
- For some functions, like \( y = 1/x \), as \( x \to +\infty \), \( y \) approaches 0. As \( x \to -\infty \), \( y \) also approaches 0.
- In other functions, like \( y = x^2 \), as \( x \to \pm \infty \), \( y \to +\infty \) (since squaring makes everything positive).
  
### **Graph Examples:**

1. **Linear Function \( y = x \)**:
   - As \( x \to +\infty \), \( y \to +\infty \).
   - As \( x \to -\infty \), \( y \to -\infty \).
   - The line goes upward to the right and downward to the left.

2. **Quadratic Function \( y = x^2 \)**:
   - As \( x \to +\infty \), \( y \to +\infty \).
   - As \( x \to -\infty \), \( y \to +\infty \).
   - The graph forms a parabola that opens upwards, stretching towards infinity as \( x \) moves away from 0.

### Why It Happens:
- **Infinity** represents an unbounded value, so as you move along the graph, the curve either increases or decreases without bound.
- Functions like \( y = x^2 \) will always tend toward infinity, because squaring any real number (positive or negative) gives a positive value.
- For rational functions like \( y = 1/x \), the behavior is different; the function tends toward 0, since dividing by large numbers (positive or negative) yields smaller and smaller values.

Would you like a visual graph representation for these examples?

Explore how graphs behave as values approach negative infinity (-INF) and positive infinity (+INF). This detailed explanation covers key functions like y = 1/x, y = x, and y = x^2, illustrating the concepts of limits and end behavior of functions. Suitable for high school students in Grades 10-12.

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