Part A: Two types of transformations can be used to transform f(x) to g(x): vertical translation and horizontal translation.
Part B: Vertical translation k = 2.
Horizontal translation k = -3
Part C: The amount of translation is represented by the constant k in the equations g(x) = f(x) + k (vertical) and g(x) = f(x - k) (horizontal).
Part A: Two types of transformations that can be used to transform f(x) to g(x) are:
1. Vertical translation: This transformation moves the graph of f(x) up or down without changing its slope or direction. In the case of the graph in the image, g(x) is a vertical translation of f(x) upward by 2 units. This means that for any input value x, the output value of g(x) is 2 units more than the output value of f(x).
2. Horizontal translation: This transformation moves the graph of f(x) left or right without changing its slope or direction. In the case of the graph in the image, g(x) is a horizontal translation of f(x) to the right by 3 units. This means that for any input value x, the output value of g(x) is the same as the output value of f(x) evaluated at x - 3.
Part B: Solving for k in each type of transformation:
1. Vertical translation: The amount of vertical translation is represented by the constant k in the equation g(x) = f(x) + k. In the case of the graph in the image, k = 2.
2. Horizontal translation: The amount of horizontal translation is represented by the constant k in the equation g(x) = f(x - k). In the case of the graph in the image, k = -3 (note that the translation is to the right, so k is negative).
Part C: Writing equations for each type of transformation:
1. Vertical translation: The equation for g(x) is g(x) = f(x) + 2.
2. Horizontal translation: The equation for g(x) is g(x) = f(x + 3).