Hello there! As a mathematics educator with a passion for clarity and precision, I'm here to help you understand the concept of reflection over the y-axis and how it relates to other lines of reflection.

Reflection in geometry is a transformation that flips a shape or point across a line or a plane. When we talk about reflecting over the y-axis, we're essentially talking about a mirror image that is symmetrical with respect to the y-axis.

To reflect a point over the y-axis, you simply take the x-coordinate of the point and change its sign. This is because the y-axis acts as a mirror, and the left side of the y-axis is a mirror image of the right side. So, if you have a point \( P(x, y) \), its reflection over the y-axis would be \( P'(-x, y) \). This means that the y-coordinate remains the same, but the x-coordinate is negated.

Now, let's delve into reflecting over the line \( y = x \) and \( y = -x \). The line \( y = x \) is a diagonal line that bisects the first and third quadrants of the Cartesian plane. Reflecting over this line means that the x and y coordinates of a point switch places. So, if you have a point \( P(x, y) \), its reflection over the line \( y = x \) would be \( P'(y, x) \).

On the other hand, the line \( y = -x \) is a diagonal line that bisects the second and fourth quadrants. Reflecting over this line not only switches the x and y coordinates but also negates both, effectively changing the signs of both coordinates. So, if you have a point \( P(x, y) \), its reflection over the line \( y = -x \) would be \( P'(-y, -x) \).

It's important to note that these reflections are not just limited to points. They can also be applied to entire shapes and figures. When a shape is reflected over the y-axis, every point in the shape undergoes the same transformation, resulting in a mirrored image that is symmetrical with respect to the y-axis.

In summary, reflecting over the y-axis involves negating the x-coordinate of a point, while reflecting over the lines \( y = x \) and \( y = -x \) involves switching and negating the coordinates, respectively. These transformations are fundamental to understanding symmetry and can be applied to a wide range of mathematical and geometric problems.

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the y-axis is the point (-x,y). Reflect over the y = x: When you reflect a point across the line y = x, the x-coordinate and y-coordinate change places. If you reflect over the line y = -x, the x-coordinate and y-coordinate change places and are negated (the signs are changed).read more >>