Question

describe how this equation y=\frac{1}{2}x^{2}, changes to parent graph

Answer 1

The equation \( y = \frac{1}{2}x^2 \) represents a transformation of the parent graph of the quadratic function \( y = x^2 \). Here's how this equation differs from the parent graph:

1. **Vertical Stretch/Compression**:
- The coefficient \( \frac{1}{2} \) in front of \( x^2 \) indicates a vertical compression by a factor of \( \frac{1}{2} \). This means that the graph of \( y = \frac{1}{2}x^2 \) will be "flatter" compared to the parent graph \( y = x^2 \). Specifically, for any value of \( x \), the corresponding \( y \) value in the new graph will be half of what it would be in the parent graph.

2. **Vertex**:
- Both the parent graph \( y = x^2 \) and the transformed graph \( y = \frac{1}{2}x^2 \) have their vertex at the origin \( (0, 0) \). There is no horizontal or vertical shift in this case.

3. **Shape of the Graph**:
- The overall shape of the graph remains a parabola opening upwards, but due to the compression, the width of the parabola increases. It will take longer to reach the same \( y \) values as \( y = x^2 \) for larger \( |x| \) values.

### Comparison:
- At \( x = 1 \):
- \( y = x^2 \) gives \( y = 1 \).
- \( y = \frac{1}{2}x^2 \) gives \( y = \frac{1}{2} \).
- At \( x = 2 \):
- \( y = x^2 \) gives \( y = 4 \).
- \( y = \frac{1}{2}x^2 \) gives \( y = 2 \).

In summary, \( y = \frac{1}{2}x^2 \) is a vertical compression of the parent quadratic function \( y = x^2 \), maintaining the same vertex at the origin and preserving the upward opening shape of the parabola.