The function g (x) = x^2+3. The function f (x) = g (x+2). The function f (x) is shifted horizontally how many places to where? The function f (x) is shifted vertically how many places where?

The function f (x) is shifted 2 units to the left and 3 units up

Step-by-step explanation:

* Lets revise the translation of a function

- If the function f (x) translated horizontally to the right

by h units, then the new function g (x) = f (x - h)

- If the function f (x) translated horizontally to the left

by h units, then the new function g (x) = f (x + h)

- If the function f (x) translated vertically up

by k units, then the new function g (x) = f (x) + k

- If the function f (x) translated vertically down

by k units, then the new function g (x) = f (x) - k

* Lets solve the problem

∵ g (x) = x² + 3

∵ f (x) = g (x + 2)

- Lets find g (x + 2) by replacing x in g (x) by (x + 2)

∵ g (x) = x² + 3

∴ g (x + 2) = (x + 2) ² + 3

∵ f (x) = g (x + 2)

∴ f (x) = (x + 2) ² + 3

- The parent function of f (x) is x²

∵ x² changed to (x + 2) ²

- Use the rules above

∴ f (x) is translated 2 units to the left

∵ (x + 2) ² added by 3

- Use the rules above

∴ f (x) is translated 3 units up

* The function f (x) is shifted 2 units to the left and 3 units up