6. Transformations

Lesson

A reflection is what 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 be the same distance from the reflection line.

Have a quick play with this interactive. Here you can change the shape of the object and the position of the mirror line.

Which image below represents the reflection for the following shape about the vertical mirror line?

- ABCDABCD

Which of the following diagrams show a reflection across the given line? Select all the correct options.

- ABCDEABCDE

As we saw above, a reflection occurs when we flip an object or shape across a line like a mirror. We can reflect points, lines, or polygons on a graph by flipping them across an axis or another line in the plane.

Reflecting over the $y$`y`-axis

Note how the point $\left(-2,1\right)$(−2,1) becomes $\left(2,1\right)$(2,1). The $y$`y` -value has stayed the same while the $x$`x` -value has changed signs.In this diagram, the image is reflected across $y$`y` -axis.

Similarly the point $\left(-6,3\right)$(−6,3) becomes $\left(6,3\right)$(6,3). The $y$`y` -value have stayed the same and the $x$`x` -value has changed signs.

Reflecting over the $x$`x`-axis

Note how the point $\left(4,3\right)$(4,3) becomes $\left(4,-3\right)$(4,−3). The $x$`x` -value has stayed the same and the $y$`y` -value has changed signs.

Similarly, the point $\left(0,5\right)$(0,5) becomes $\left(0,-5\right)$(0,−5). The $x$`x` -value have stayed the same and the $y$`y` -values has changed signs.

Summary

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

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

Plot the following.

Plot the point $A$

`A`$\left(2,-2\right)$(2,−2).Loading Graph...Now plot point $A'$

`A`′, which is a reflection of point $A$`A`about the $x$`x`-axis.Loading Graph...

Plot the following.

Plot the line segment $AB$

`A``B`, where the endpoints are $A$`A`$\left(-6,-1\right)$(−6,−1) and $B$`B`$\left(10,8\right)$(10,8).Loading Graph...Now plot the reflection of the line segment about the $y$

`y`-axis.Loading Graph...

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.