To analyze the function \( f(x) = \frac{1}{2} (x + 7)^2 + 4 \) and determine the transformations from the parent function \( g(x) = x^2 \), we can break it down into its components:
-
Horizontal Shifts:
- The term \((x + 7)\) indicates a shift to the left by 7 units. This is because the function translates left when the input \(x\) is replaced with \(x + 7\).
-
Vertical Shifts:
- The \(+ 4\) outside the squared term means the graph shifts vertically upwards by 4 units.
-
Vertical Compression:
- The coefficient \(\frac{1}{2}\) in front of the squared term indicates a vertical compression by a factor of 2 (the graph stretches wider compared to the parent function).
-
Reflection:
- There is no negative sign in front of the \(\frac{1}{2}\), so there is no reflection across the x-axis.
Putting it all together, the transformations are:
- Shift left 7 units
- Shift up 4 units
- Vertical compression by a factor of 2
Hence, the correct transformations from the given options are:
- left 7 units
- up 4 units
- vertical compression
The other options, such as down 7 units, down 4 units, right shifts, and reflections, do not apply to this function.