1. The problem gives two functions: $$f(x) = x^2$$ and $$g(x) = (x - 9)^2 + 3$$. 2. We are asked to understand or analyze these functions. Both are quadratic functions, which means their graphs are parabolas. 3. The general form of a quadratic function is $$y = (x - h)^2 + k$$, where $h$ and $k$ represent the vertex coordinates. 4. For $$f(x) = x^2$$, the vertex is at $(0,0)$ because it can be written as $$(x - 0)^2 + 0$$. 5. For $$g(x) = (x - 9)^2 + 3$$, the vertex is at $(9,3)$. 6. This means $$g(x)$$ is the graph of $$f(x)$$ shifted 9 units to the right and 3 units up. 7. Both parabolas open upwards because the coefficient of the squared term is positive. 8. The function $$f(x)$$ has its minimum value at 0, and $$g(x)$$ has its minimum value at 3. Final answer: $$f(x) = x^2$$ is a parabola with vertex at $(0,0)$, and $$g(x) = (x - 9)^2 + 3$$ is the same parabola shifted right by 9 and up by 3, with vertex at $(9,3)$.