Final answer:
The point (4, -4) flipped over the y-axis becomes (-4, -4). This is achieved by reversing the sign of the x-coordinate while keeping the y-coordinate unchanged. In vector terms, we invert the sign of the x-component while the magnitude remains the same.
Step-by-step explanation:
Flipping a point over the y-axis changes the sign of the x-coordinate while the y-coordinate remains the same. If we take the point (4, -4) and flip it over the y-axis, we follow a procedure that mirrors the point to the opposite side of this axis. The original point has an x-coordinate of 4, so the flipped point will have an x-coordinate of -4, as flipping over the y-axis means reversing the direction on the horizontal axis. The y-coordinate remains unchanged, so the point (4, -4) flipped over the y-axis becomes (-4, -4).
In terms of vector notation and considering a two-dimensional Cartesian coordinate system with the unit vectors î and ĵ, where î denotes the direction along the x-axis and ĵ along the y-axis, we can represent the flip algebraically. The vector x-component Ďx = −4.0î of the displacement vector has the magnitude |Ďx| = 4.0 because the magnitude of the unit vector is |î| = 1. Inverting this vector, we write Ďx' = 4.0î. Therefore, flipping the point over the y-axis is represented by changing the sign of the vector's x-component.
Reflecting points over the y-axis is also related to the concept of even and odd functions in mathematics, where an even function is symmetric about the y-axis. By definition, for an even function: y(x) = y(-x). Reflecting a point over the y-axis embodies this concept as the x-coordinates have their signs reversed, resulting in the point being on the opposite side of the y-axis but at the same distance from it.