New Zealand
Level 6 - NCEA Level 1

# Reflections on Cartesian Plane

Lesson

## Reflections

As we saw in the previous lesson on Flips, A reflection occurs when we flip an object or shape across a line. Like a mirror, the object is exactly the same size, just flipped in position.  So what was on the left may now appear on the right.  Every point on the object or shape has a corresponding point on the image, and they will both have the same distance from the reflection line.  We can reflect points, lines, polygons on the Cartesian plane by flipping them across an axis line or another line in the plane.

If we reflect horizontally across the $y$y axis, then the y values of the coordinates remain the same and the $x$x values change sign.

In this diagram, the image is reflected across $y$y axis.

Note how the point $\left(-2,1\right)$(2,1) becomes $\left(2,1\right)$(2,1).  The $y$y values have not changed and the x values have changed signs.

Similarly the point $\left(-6,3\right)$(6,3) becomes $\left(6,3\right)$(6,3).  The $y$y values have not changed and the x values have changed signs.

If we reflect vertically across the $x$x axis, then the $x$x values of the coordinates remain the same and the $y$y values change sign.

In this diagram, the image is reflected across $x$x axis.

Note how the point $\left(4,3\right)$(4,3) becomes $\left(4,-3\right)$(4,3).  The $x$x values have not changed and the $y$y values have changed signs.

Similarly the point $\left(0,5\right)$(0,5) becomes $\left(0,-5\right)$(0,5).  The $x$x values have not changed and the $y$y values have changed signs.

Have a quick play with this interactive to further consolidate the ideas behind translations on the Cartesian plane.

Let's have a look at these worked examples.

##### Question 1

Consider the point $A\left(7,3\right)$A(7,3).

1. Plot point $A$A on the number plane.

2. Now plot point $A'$A, a reflection of point $A$A across the $x$x-axis.

##### Question 2

Consider the point $A\left(-7,-3\right)$A(7,3).

1. Plot point $A$A on the number plane.

2. Now plot point $A'$A, a reflection of point $A$A across the $y$y-axis.

##### Question 3

Consider the line segment $AB$AB, where the endpoints are $A$A$\left(-4,-2\right)$(4,2) and $B$B$\left(6,7\right)$(6,7).

1. Plot the line segment $AB$AB on the number plane.

2. Now plot the reflection of the line segment $AB$AB across the $x$x-axis.

##### Question 4

Plot the new triangle formed by reflecting the given triangle about the line $x=-3$x=3.

### Outcomes

#### GM6-8

Compare and apply single and multiple transformations

#### GM6-9

Analyse symmetrical patterns by the transformations used to create them

#### 91034

Apply transformation geometry in solving problems