Final answer:
The function g(x) is derived from f(x) through a horizontal translation 5 units to the left and a vertical translation 2 units upwards. This changes the position of f(x) on the coordinate plane without altering its shape.
Step-by-step explanation:
The transformation described by the student in the function g(x) = f(x + 5) + 2 involves two main operations: a horizontal translation and a vertical translation. To understand this, let's break down the components of the transformation.
Horizontal Translation
When we see f(x + 5), it suggests a horizontal shift of the function f(x). As per the algebraic rules, adding a number inside the function's argument (x + 5) shifts the graph in the opposite direction of the sign. So, in this case, it translates the graph of f(x) 5 units to the left.
Vertical Translation
The + 2 on the outside of the function means that after its horizontal shift, the graph is then moved upwards by 2 units. This is a vertical shift that raises every point of the function f(x) by 2 units in the positive y-direction.
Together, these transformations would result in the function g(x) being the graph of f(x), shifted 5 units left and 2 units up.