Transform y = x² : Shifting 2 Right and 5 Up - Finding the Equation

To solve this problem, we'll start by understanding the transformations required:

• The original function is $y = x^2$.
• We need to move this function 2 spaces to the right and 5 spaces upwards.

Step 1: Apply the horizontal shift 2 units to the right.
To shift a function horizontally, replace $x$ with $x - h$, where $h$ is the shift to the right. Thus, we replace $x$ with $x - 2$ to get:

$y = (x - 2)^2$.

Step 2: Apply the vertical shift 5 units upwards.
To shift a function vertically, add $k$ to the function, where $k$ is the number of units to shift up. Thus:

$y = (x - 2)^2 + 5$.

Combining these transformations, the equation of the transformed function is:

$y = (x - 2)^2 + 5$.

This matches the choice labeled as 3. Thus, the correct equation after translating the parabola 2 spaces to the right and 5 spaces upwards is:

$y = (x - 2)^2 + 5$.