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

example 2 – simplify: step 1: completely factor both the numerators and denominators of all fractions. step 2: change the division sign to a multiplication sign and flip (or reciprocate) the fraction after the division sign; essential you need to multiply by the reciprocal. step 3 : cancel or reduce the fractions.