The image you uploaded contains a function $f(x) = -0.5x^2 + 1$ and a partially filled-in chart with graphing details, such as domain, range, intercepts, and increasing/decreasing intervals. Below, I'll walk through the problem step-by-step, providing explanations for each component of the function and the requested details.

1. Parent Function:

The given function is a quadratic function $f(x) = -0.5x^2 + 1$, so its parent function is: $f(x) = x^2$ This is a standard parabola that opens upward, with the vertex at $(0, 0)$.

2. Transformations:

There are two transformations applied to the parent function:

• Vertical Compression: The coefficient $-0.5$ compresses the graph vertically by a factor of 0.5 and reflects it over the x-axis.
• Vertical Shift: The constant term $+1$ shifts the graph upward by 1 unit.

3. Graphing:

The graph is a downward-facing parabola due to the negative coefficient. The vertex of the parabola is at $(0, 1)$, which comes from the transformation of the parent function.

4. Domain:

The domain of a quadratic function is all real numbers: [ \text