Question: Given $f(x) = x^2$, after performing the following transformations: shift upward 52 units and shift 97 units to the right, the new function $g(x) = __________$
1. **State the problem:** We start with the function $f(x) = x^2$ and apply two transformations: shift upward by 52 units and shift to the right by 97 units.
2. **Recall the transformation rules:**
- Shifting a function $f(x)$ to the right by $h$ units results in $f(x - h)$.
- Shifting a function upward by $k$ units results in $f(x) + k$.
3. **Apply the horizontal shift:**
Shifting $f(x) = x^2$ to the right by 97 units gives:
$$f(x - 97) = (x - 97)^2$$
4. **Apply the vertical shift:**
Shifting the function upward by 52 units gives:
$$g(x) = (x - 97)^2 + 52$$
5. **Final function:**
$$g(x) = (x - 97)^2 + 52$$
This is the new function after the given transformations.