How to Reflect a Shape in y = −x Using Cartesian Coordinates
Reflecting a Shape in y = −x Using Cartesian Coordinates
A shape can be reflected in the line y = −x.
If point on a shape is reflected in the line y = −x:

both coordinates change sign (the coordinate becomes negative if it is positive and vice versa)

the xcoordinate becomes the ycoordinate and the ycoordinate becomes the xcoordinate
The image below shows a point on a shape being reflected in the line y = −x:

The point A has Cartesian coordinates (−3, 5).

The reflected point A' has Cartesian coordinates (−5, 3).
The xcoordinate of A (−3) has its sign changed (3) and becomes the ycoordinate of A'. The ycoordinate of A (5) has its sign changed (−5) and becomes the xcoordinate of A'.
How to Reflect a Shape in y = −x Using Cartesian Coordinates
Reflecting a shape in the line y = −x using Cartesian coordinates is easy.
Question
Reflect the shape below in the line y = −x.
StepbyStep:
1
Find the Cartesian coordinates of each point on the shape.
Write the xcoordinates and ycoordinates of each point.
Point  xcoordinate  ycoordinate 

A  −6  3 
B  −3  2 
C  −6  1 
2
Change the sign of both coordinates.
Make them negative if they are positive and positive if they are negative.
Point  xcoordinate  ycoordinate  −xcoordinate  −ycoordinate 

A  −6  3  6  −3 
B  −3  2  3  −2 
C  −6  1  6  −1 
3
Using the new coordinates from Step 2 (that have had their signs changed), make the xcoordinate the ycoordinate and the ycoordinate the xcoordinate.
−xcoordinate  −ycoordinate  Point  xcoordinate  ycoordinate 

6  −3  A'  −3  6 
3  −2  B'  −2  3 
6  −1  C'  −1  6 
4
Plot the reflected points and draw in the shape.
Answer:
We have reflected the shape in the line y = −x.
A Formula to Reflect a Point in y = −x Using Cartesian Coordinates
In general, we write Cartesian coordinates as:
x is the xcoordinate. y is the ycoordinate. x and y can taken any number.
The reflected point has Cartesian coordinates:
The image below shows a general Cartesian coordinate being reflected in the line y = −x: