Find the coordinates of the image of the point p(4,6)under the reflection about the line x=3 followed by the reflection about the line y=x.

reflect through x=3: (x,y)->(3-(x-3),y) = (6-x,y)

reflect through y=x: (x,y)->(y,x)

so, now just combine those two transformations:

(x,y)->(6-x,y)->(y,6-x)

To find the coordinates of the image of point P(4, 6) under the reflection about the line x = 3 followed by the reflection about the line y = x, follow these steps:

Step 1: Reflection about the line x = 3
To reflect a point about the line x = 3, we need to find the distance between the point and the line, and then find the point on the other side of the line with the same distance.

The distance between the point P(4, 6) and the line x = 3 is 4 - 3 = 1. So, the distance of the image point from the line x = 3 will also be 1. Since the line x = 3 is a vertical line, the x-coordinate of the image point will be 3, and the y-coordinate will be 6 + 2(1) = 8.

So, after the reflection about the line x = 3, the new point becomes Q(3, 8).

Step 2: Reflection about the line y = x
To reflect the point Q(3, 8) about the line y = x, we need to swap the x and y coordinates.

The new point will have the x-coordinate equal to the y-coordinate of Q, and the y-coordinate equal to the x-coordinate of Q.

So, after the reflection about the line y = x, the new point becomes R(8, 3).

Therefore, the coordinates of the image of point P(4, 6) under the reflection about the line x=3 followed by the reflection about the line y=x are R(8, 3).

To find the coordinates of the image of point P(4, 6) under the given sequence of reflections, we can follow these steps:

1. Reflection about the line x = 3:
- Since the line of reflection is vertical, the x-coordinate of the point remains the same, but the y-coordinate changes.
- To find the new y-coordinate, we can calculate the distance between the original point and the line of reflection (x = 3) and then mirror that distance on the other side of the line.
- The distance between the point P(4, 6) and the line x = 3 is |4 - 3| = 1.
- The new y-coordinate will be the original y-coordinate minus twice the distance calculated above.
- The new y-coordinate = 6 - 2(1) = 6 - 2 = 4.
- So, after the first reflection, the image of point P(4, 6) under the line x = 3 is Q(4, 4).

2. Reflection about the line y = x:
- In this reflection, both the x and y coordinates are swapped.
- So, the new x-coordinate will be the previous y-coordinate, and the new y-coordinate will be the previous x-coordinate.
- Applying this rule, the new coordinates of point Q(4, 4) become R(4, 4).

Therefore, the coordinates of the image of point P(4, 6) under the given sequence of reflections are R(4, 4).