To translate the function y = x 2 units right and 3 units up, we need to replace x with x - 2 and add 3 to y. This is because shifting x by 2 units changes the horizontal position of the graph, and adding 3 to y changes the vertical position of the graph. The translated function is:
y + 3 = (x - 2)
To write this function in vertex form, we need to complete the square on the right side of the equation. This means adding and subtracting the same number inside the parentheses to make a perfect square. The number we need to add and subtract is half of the coefficient of x, squared. In this case, the coefficient of x is -1, so half of it is -1/2, and squaring it gives 1/4. The equation becomes:
y + 3 = (x - 2) + 1/4 - 1/4
Now we can factor the first three terms as a perfect square using the identity (x + a)^2 = x^2 + 2ax + a^2. In this case, x + a = x - 1/2, so we get:
y + 3 = (x - 1/2)^2 - 1/4
Finally, we can rearrange the equation to get y on one side and simplify the constants. We get:
y = (x - 1/2)^2 - 13/4
This is the equation of the function in vertex form. The vertex is (h, k) = (1/2, -13/4).