The function f(x) = x2. the graph of g(x) is f(x) translated to the left 6 units and down 5 units. what is the function rule for g(x)?
a. g(x) = (x + 6)2 - 5
b. g(x) = (x - 5)2 + 6
c. g(x) = (x - 6)2 + 5
d. g(x) = (x + 5)2 - 6

   The function rule for g(x) = f(x) shifted left by 6 units and down by 5 units. To shift a function up or down, you add or subtract the modified function by the amount that you're shifting up or down. For instance, x^2 moved 3 units down would be x^2-3. If you're moving a function to the left or right, you will add or subtract the "x" term of the function such that if you're moving to the left, you'll be adding the amount you moved and if you're shifting to the right, you'll be subtracting the amount you moved. For example, f(x)= x^2, let's say, and this function is shifted two units to the left. First, you would take the input of "x" before the "f of" is applied to it, and you'd add two to x, as it is being shifted to the right by 2 units. Your resulting function, after applying the function to the modified input, would be (x+2)^2. this applies to the problem that you're mentioning. You have a composition of functions with translations, so you just take everything step by step. First, deal with modifying the input such that the amount shifted left or right is accommodated for. Next, add or subtract the amount that you shifted up or down (vertically as opposed to horizontally) to the already modified function of f(x). All of the steps combined together would give you g(x). I hope this helped you! :)