The problem provided in the image asks you to perform transformations on the function $f(x) = x^2$. Specifically:

1. Shift the function upward by 34 units.
2. Shift the function 59 units to the right.

To perform these transformations, we modify the function $f(x)$ as follows:

1. Vertical Shift (upward by 34 units):

To shift a function upward, we add the desired number of units to the output. So, for the vertical shift of 34 units:

$f(x) = x^2 \quad \text{becomes} \quad x^2 + 34$

2. Horizontal Shift (to the right by 59 units):

To shift a function to the right, we replace $x$ with $(x - h)$, where $h$ is the number of units to shift. Since we are shifting 59 units to the right:

$f(x) = x^2 \quad \text{becomes} \quad (x - 59)^2$

Combining Both Transformations:

We now combine both the horizontal and vertical shifts:

$g(x) = (x - 59)^2 + 34$

Thus, the new function is:

$g(x) = (x - 59)^2 + 34$

Would you like a detailed breakdown of how these transformations work or have any further questions?

Here are 5 related questions you could explore:

1. How do horizontal shifts differ from vertical shifts in function transformations?
2. What is the effect of shifting a function downward instead of upward?
3. How would you shift the function $f(x) = x^2$ left by 10 units and downward by 5 units?
4. What happens to the graph of $f(x) = x^2$ when reflected across the y-axis?
5. How does the vertex of a parabola change under horizontal and vertical shifts?

Tip: For horizontal shifts, remember to adjust the variable $x$ inside the function, and for vertical shifts, adjust the entire function's output directly.