Step-by-step explanation:
g(x) has clearly in its core the same function curve.
but it is
1. upside-down
2. moved up from the x-axis by 1 unit
3. moved to the right of the y-axis by 3 units
so, how do we express these 3 attributes in the functional definition ?
1. easily : by flipping the sign. g(x) = -x² is the same function type just upside-down (mirrored into the negative y space).
g(x) = -x²
2. also very easy : a function is moved up or down on the coordinate grid by adding (or subtracting) a constant.
we need to move our function up by 1 unit : we add 1.
that makes currently g(x) = -x² + 1
3. this is the trickiest part. to move a function left or right we need to make the function "think" that the input value x is not x, but it is (x ± constant).
let's ignore 1. and 2. for the moment and just focus moving the original function 3 units to the right.
that tells us that the functional result value of x in the shifted function must be the same as the functional result value in the original function for an input value that is 3 units "earlier" on the x-axis.
that would mean g(0) = f(-3), g(1) = f(-2), g(2) = f(-1), g(3) = f(0), ...
so, we see, g(x) = f(x-3) = (x-3)²
now, we combine again 1., 2. and 3., and we get
g(x) = -f(x-3) + 1 = -(x-3)² + 1
