Final answer:
The correct translation that maps f(x) = x² onto g(x) = x² + 2x + 6 is left 1 unit, up 5 units, which corresponds to answer choice (A). This was determined by rewriting g(x) in vertex form as (x + 1)² + 5.
Step-by-step explanation:
To determine which translation maps the graph of the function f(x) = x² onto the function g(x) = x² + 2x + 6, we need to look at how the second equation differs from the first. The quadratic function g(x) can be rewritten in vertex form to identify the translation easily. Completing the square for the equation g(x) will give us:
g(x) = x² + 2x + 1 + 5 = (x + 1)² + 5
The term (x + 1)² indicates a horizontal shift to the left by 1 unit, and the +5 indicates a vertical shift upwards by 5 units. Thus, the correct translation is left 1 unit, up 5 units. So the correct answer to the question is (A) Left 1 unit, up 5 units.
Remember from your study of algebra that if f(x) is some function, then f(x - d) is the same function translated in the positive x-direction by a distance d. The function f(x + d) is the same function translated in the negative x-direction by a distance d.