Question

reasoning one image of $\triangle abc$ is $\triangle abc$ how do the x coordinates of the vertices change? how do the y - coordinates of the vertices change? what type of reflection is the image $\triangle abc$?
how do the x - coordinates of the vertices change?
a. the x - coordinates of the vertices are unchanged in the image
b. the x - coordinates of the vertices are the same distance away from the y - axis but in the opposite direction
c. the x - coordinates of the vertices are the same distance away from the x - axis but in the opposite direction
d. the x - coordinates of the vertices change differently depending on where they are on the figure
how do the y - coordinates of the vertices change?
a. the y - coordinates of the vertices are unchanged in the image
b. the y - coordinates of the vertices are the same distance away from the y - axis but in the opposite direction
c. the y - coordinates of the vertices change differently depending on where they are on the figure
d. the y - coordinates of the vertices are the same distance away from the x - axis but in the opposite direction

Explanation:

For the x - coordinate change question:

When a figure is reflected over the y - axis, the rule for the transformation of coordinates \((x,y)\) is \((-x,y)\). This means that the x - coordinate of each vertex is the same distance from the y - axis (since the distance from a point \((x,y)\) to the y - axis is \(|x|\), and for \((-x,y)\) it is \(|-x|=|x|\)) but in the opposite direction (positive x becomes negative and vice - versa). Option A is wrong because x - coordinates do change. Option C is wrong because the change in x - coordinates follows a specific rule (reflection over y - axis) and is not related to distance from x - axis. Option D is wrong as the change in x - coordinates is uniform (follows the reflection over y - axis rule) and does not depend on the position in the figure.

When reflecting over the y - axis, the y - coordinate of a point \((x,y)\) remains the same. So the y - coordinates of the vertices are unchanged in the image. Option B is wrong because it describes a change related to y - axis distance which is not the case for y - coordinates in a y - axis reflection. Option C is wrong as y - coordinates do not change differently based on position. Option D is wrong as it describes a change related to x - axis distance which is not relevant for y - coordinates in a y - axis reflection.

Answer:

B. The x - coordinates of the vertices are the same distance away from the y - axis but in the opposite direction

For the y - coordinate change question: