Final answer:
The correct transformation of the graph of f(x) = (x - 4)^2 that will coincide with the graph of g(x) = 3(x + 1)^2 is D: g(x) = {f(x + 5), which translates f(x) 5 units to the left.
Step-by-step explanation:
When we look at the transformations of functions in mathematics, particularly when comparing the functions f(x) = (x - 4)^2 and g(x) = 3(x + 1)^2, we are looking for specific shifts, stretches, or reflections that will align one graph with another. The function g(x) can be derived from f(x) by applying a series of transformations.
To find which option coincides with the graph of g(x), we should match the transformations with the changes seen in g(x) compared to f(x). Based on the given options:
- Option A: g(x) = f(x - 5) translates f(x) 5 units to the right.
- Option B: g(x) = 6f(x + 5) translates f(x) 5 units to the left and stretches it vertically by a factor of 6.
- Option C: g(x) = 6f(x - 5) translates f(x) 5 units to the right and stretches it vertically by a factor of 6.
- Option D: g(x) = {f(x + 5) translates f(x) 5 units to the left.
By examining the given function g(x), we see it involves a horizontal translation of 5 units to the left (since x+1 is equivalent to x-(-1)) and a vertical stretch by a factor of 3. Therefore, the correct option that performs these transformations on f(x) to coincide with g(x) is Option D: g(x) = {f(x + 5).