The equation \( y^2 - x^2 = 1 \) is a form of a hyperbola equation. This specific form is often called the standard form of a **rectangular hyperbola** oriented along the axes.

Here’s how we can analyze it:

### 1. Rearranging the equation:
\[
y^2 - x^2 = 1 \quad \text{can be rewritten as} \quad \frac{y^2}{1} - \frac{x^2}{1} = 1
\]
This is the standard form of a hyperbola:
\[
\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1
\]
Where:
- \( a = 1 \), meaning the distance between the center and the vertices along the \( y \)-axis is 1.
- \( b = 1 \), meaning the distance between the center and the co-vertices along the \( x \)-axis is also 1.

### 2. Characteristics of the hyperbola:
- **Center**: The center of the hyperbola is at \( (0, 0) \), since the equation is in standard form without shifts.
- **Vertices**: The vertices are at \( (0, \pm a) = (0, \pm 1) \), which are points on the y-axis.
- **Asymptotes**: The asymptotes are straight lines that the hyperbola approaches but never touches. For this equation, the asymptotes are the lines:
  \[
  y = x \quad \text{and} \quad y = -x
  \]
- **Axes**: The transverse axis (where the vertices lie) is the \( y \)-axis, and the conjugate axis is the \( x \)-axis.
  
### 3. Graph of the Hyperbola:
This is a hyperbola that opens upwards and downwards, centered at the origin with its asymptotes intersecting at the origin.

Would you like a visual representation of this hyperbola?

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**Related Questions:**
1. How do you graph a hyperbola given its equation?
2. What is the difference between a hyperbola and an ellipse?
3. How do you find the foci of a hyperbola?
4. How do the asymptotes of a hyperbola affect its shape?
5. What happens if the equation has a different constant, like \( y^2 - x^2 = 4 \)?

**Tip:** In hyperbolas, the distance from the center to the foci can be found using the relation \( c^2 = a^2 + b^2 \).

Analyze the hyperbola represented by the equation y² - x² = 1. Learn its properties, including center, vertices, and asymptotes, and explore how to graph it. Suitable for students in Grades 10-12, this solution covers key algebraic concepts related to conic sections and hyperbolas.

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