The given function is $f(x) = x^2$.

You are asked to apply two transformations to this function:

1. Shift it upward by 20 units.
2. Shift it to the right by 51 units.

Step-by-Step Solution

1. Upward Shift: To shift $f(x)$ upward by 20 units, we add 20 to the function: $f(x) + 20 = x^2 + 20$

2. Rightward Shift: To shift the function to the right by 51 units, replace $x$ with $x - 51$ in the function: $f(x - 51) = (x - 51)^2$

3. Combine Both Transformations: Apply both transformations to get the new function $g(x)$: $g(x) = (x - 51)^2 + 20$

Final Answer

$g(x) = (x - 51)^2 + 20$

Would you like further details on how each transformation affects the graph of the function? Here are some related questions to explore:

1. What effect does a vertical shift have on the function's graph?
2. How does shifting to the right differ from shifting to the left?
3. Can we combine multiple transformations in any order, or does the order matter?
4. How would this transformation look for a different base function, like $f(x) = x^3$?
5. What happens to the function if instead of shifting up, we shift down by 20 units?

Tip: Remember that shifting horizontally involves changes within the parentheses with $x$, while vertical shifts are added or subtracted outside the function.