The equation you provided, \( y = 1 - x^2 \), represents a parabola. Here's an analysis of this function:

### 1. General Form
This equation can be seen as a quadratic function in the form:
\[
y = -x^2 + 1
\]
or equivalently,
\[
y = -x^2 + c
\]
where \( c = 1 \) is the constant term.

### 2. Key Properties of \( y = 1 - x^2 \)
1. **Vertex**: Since the parabola opens downwards (due to the negative coefficient of \( x^2 \)), the vertex will be the highest point of the parabola. The vertex is at \( (0, 1) \), found by setting \( x = 0 \) to maximize \( y \).
   
2. **Axis of Symmetry**: The axis of symmetry is the line \( x = 0 \), as the parabola is symmetric about the y-axis.

3. **Direction**: The parabola opens **downwards** because the coefficient of \( x^2 \) is negative.

4. **Intercepts**:
   - **Y-intercept**: This is the value of \( y \) when \( x = 0 \). Here, \( y = 1 \), so the y-intercept is at \( (0, 1) \).
   - **X-intercepts**: These occur when \( y = 0 \). Solving \( 0 = 1 - x^2 \) gives:
     \[
     x^2 = 1 \Rightarrow x = \pm 1
     \]
     So, the x-intercepts are at \( (1, 0) \) and \( (-1, 0) \).

### 3. Graph Shape
The graph of \( y = 1 - x^2 \) is a parabola that opens downward, with its highest point at \( (0, 1) \). This parabola is symmetric about the y-axis.

### Summary
- Vertex: \( (0, 1) \)
- Axis of Symmetry: \( x = 0 \)
- Y-intercept: \( (0, 1) \)
- X-intercepts: \( (1, 0) \) and \( (-1, 0) \)

Would you like more details or have specific questions about this function?

### Follow-up Questions
1. How would you find the domain and range of this function?
2. What transformations would occur if we modified this function to \( y = 1 + x^2 \)?
3. How would the graph change if the equation were \( y = -2x^2 + 1 \)?
4. What is the importance of the vertex form for understanding parabolas?
5. How can we calculate the focus and directrix of this parabola?

#### Tip
For any quadratic function in the form \( y = ax^2 + bx + c \), the direction the parabola opens is determined by the sign of \( a \): if \( a \) is positive, it opens upwards; if \( a \) is negative, it opens downwards.

Explore the properties of the parabola defined by the equation y = 1 - x^2. This comprehensive guide covers key features such as vertex, intercepts, and symmetry, providing insights suitable for Grades 8-10 students.

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