The given function is
\[ y = \left(-\frac{1}{2}\right)(x-1)^2 - 5. \]
To identify the parent function, we can look at the structure of the function. The expression \((x-1)^2\) suggests that the basic shape of the function is a quadratic function, which is characterized by the general equation
\[ y = x^2. \]
This means that the parent function is:
Parent Function: \( y = x^2 \)
The transformations applied to the parent function to obtain the given function are:
- Horizontal Shift: The term \((x-1)\) indicates a shift to the right by 1 unit.
- Vertical Stretch/Compression: The coefficient \(-\frac{1}{2}\) indicates a vertical compression (since the absolute value is less than 1) and a reflection in the x-axis (due to the negative sign).
- Vertical Shift: The \(-5\) indicates a downward shift by 5 units.
Putting all these transformations together gives you the function described:
- Shift right by 1 unit
- Reflect across the x-axis
- Compress vertically by a factor of 2
- Shift down by 5 units
Thus, the parent function is \( y = x^2 \).