The Lesson
A shape can be
reflected in the line
y = −x.
If point on a shape is reflected in the line
y = −x:
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.
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:
Lesson Slides
The slider below shows another real example of how to reflect a shape in the line y = −x using Cartesian coordinates.
Open the slider in a new tab
What Is a Reflection?
A reflection flips a shape so that it becomes a mirror image of itself.
A reflection is a type of
transformation.