Final Answer:
The transformation from the graph of f(x) = -4x + 8 to g(x) = -f(x) corresponds to a reflection in the x-axis. This is achieved by multiplying each y-coordinate of the points on the graph of f(x) by -1. Consequently, the resulting graph g(x) is a reflection of f(x) in the x-axis.
3.reflection in the x-axis
Step-by-step explanation:
The transformation from the function f(x) = -4x + 8 to g(x) = -f(x) involves a comprehensive alteration of the graph. The function g(x) is essentially a reflection of f(x) across the x-axis. To understand this transformation, consider each point on the graph of f(x), where the x-coordinate remains unchanged, but the y-coordinate undergoes a multiplication by -1. This multiplication effectively flips the position of each point with respect to the x-axis, causing the entire graph to mirror itself vertically.
Mathematically, for any point \((x, y)\) on \(f(x)\), the corresponding point on g(x) is (x, -y). This reversal of the y-coordinates transforms the concave-upward parabolic shape of f(x) into a concave-downward parabola for g(x). The symmetry axis of the parabola also switches from the x-axis to the y-axis due to the reflection.
In essence, the reflection in the x-axis is a geometric operation that inverts the orientation of the graph, turning points above the x-axis into points below it, and vice versa. This transformation is a fundamental concept in mathematics and has broad applications in various fields, including physics, computer graphics, and engineering. In the context of these two functions, it signifies a change in the overall behavior and visual representation from f(x) to g(x).