The original function is $f(x) = x^2$. We are asked to perform two transformations:

1. Shift upward by 68 units: This adds 68 to the entire function. The transformation becomes $f(x) + 68 = x^2 + 68$.

2. Shift 69 units to the right: To shift a function to the right by $h$ units, we replace $x$ with $(x - h)$ in the function. In this case, we replace $x$ with $(x - 69)$.

Thus, the new function $g(x)$ becomes:

$g(x) = (x - 69)^2 + 68$

Let me know if you would like more details or have any questions!

Here are 5 questions related to this topic:

1. How does shifting a function to the left affect the equation?
2. What happens when you shift a function downward instead of upward?
3. Can we describe the transformations as translations on the graph of the function?
4. How do horizontal and vertical shifts interact in more complex functions?
5. How would the equation change if we shifted 69 units to the left instead of right?

Tip: Remember, shifting horizontally affects the input $x$, while vertical shifts affect the output directly.