To understand the transformations that change the graph of f(x) to g(x), we first need to analyze both functions. The function f(x) is a basic quadratic function, which is represented by a parabola that opens upwards with its vertex at the origin (0, 0).
Now, looking at g(x):
g(x) = x^2 + 32 – 7
= x^2 + 25
This means g(x) is also a quadratic function, but with some modifications to f(x).
1. **Vertical Shift**: The graph of g(x) has been shifted upwards. Specifically, since +25 is added to x^2, the entire graph of f(x) is moved up by 25 units.
2. **No Horizontal Shifts**: There are no changes to the x-term in g(x), so there is no horizontal shift. The parabola remains centered along the y-axis.
3. **No Reflections or Stretching**: The coefficient for x^2 remains as 1 in both functions, indicating that the shape and orientation of the parabola have not changed; there’s no reflection or vertical stretching involved.
In summary, the major transformation that converts the graph of f(x) to g(x) is the vertical shift of 25 units upwards.