Final answer:
The representation of a vertical shift with given function f(x) = x² and k = -3 is A. f(x) + k. The correct ranking based on the axis of symmetry for functions f(x), g(x), and h(x) is C. h(x), f(x), g(x).
Step-by-step explanation:
Given the function f(x) = x² and k = -3, the option that represents a vertical shift is A. f(x) + k. A vertical shift occurs when you add or subtract a constant from the function output, hence moving the graph up or down in the coordinate system.
For the ranking of the functions based on their axis of symmetry, we first need to recall that the axis of symmetry for a quadratic function f(x) = ax² + bx + c is given by the line x = -b/(2a). Applying this to the given functions:
- f(x) = 2x² - 8x; axis of symmetry is x = 2
- g(x) = x² - 6x + 1; axis of symmetry is x = 3
- h(x) = -2x²; axis of symmetry is x = 0
Ranking these values from least to greatest gives us the order: C. h(x), f(x), g(x).