2.2 Constructing Reflections
Now we begin to look at transformations that yield congruent images. We’ll begin with reflections and
then move into a series of transformations. A series of transformations applies more than one transformation one
at a time to a pre-image.
Reflections
Reflections are like a mirror image, or flip, of the pre-image. This results in a congruent image, meaning it
is not only the same shape but also the same size. At the 8th grade, we will usually be reflecting across the -axis or
the 1-axis. Both cases need different formulas for the coordinates, so we’ll break it down one at a time.
Reflecting Across the n-Axis
A reflection across the -axis takes points above the -axis to below the -axis and vice versa. It basically
flips the shape up and down. To do so, it utilizes these formulas:
o = 1 o = −1
Notice that the -coordinate stays the same in the image as it was in the pre-image. The only real change
is that you take the opposite 1 value. So if the 1-coordinate was positive in the pre-image, it would be negative in
the image. If it was negative in the pre-image, it would be positive in the image. Basically, if you want to apply a
reflection across the -axis, just change the sign of the 1-coordinates.
While the formulas are valid for this reflection, you can imagine how annoying it would be to memorize the
formulas for reflections across lots of different lines. Therefore, it is probably much simpler for you to think of a
reflection as folding the paper on the line of reflection and stamping the pre-image wherever it lands after the fold.
Let’s take a triangle with coordinates ^: (−6,5), j: (−4,2), X: (−2,2) and reflect it across the -axis. Since
we only need to change the sign of the 1-coordinates we end up with the image of a triangle with the points
^′: (−6, −5), j′: (−4, −2), X′: (−2, −2) if we use the formulas. If you simply think of folding the paper on the axis, notice that the blue triangle would land exactly where the green one is. It’s like the blue triangle is a stamp
and stamped down the new green triangle.
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6
A
4
B
C
2
5
6
A
Reflection
across n-axis
4
B
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C
B'
C'
2
5
2
4
A'
6
8
Reflecting Across the p-Axis
Seeing how to reflect across the -axis, what do you think will happen if you reflect across the 1-axis? As
expected, it is now the -coordinates that change, and the 1-coordinates that stay the same. We use these
formulas to reflect across the 1-axis:
o = −
1o = 1
Just change the sign of the -coordinate. Let’s look at the triangle from our previous example and reflect
it across the 1-axis. You should verify that it leads to the points ^′: 6,5, j′: 4,2, X: 2,2. However, notice that
it would again be easier to think of folding the paper on the 1-axis and stamping the green triangle from the blue.
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6
A
Reflection
across p-axis
4
B
C
2
5
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B
5
A'
4
C
2
C'
5
B'
5
2
2
4
4
6
6
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8
Reflecting Across Other Lines
Since every line has a new formula to reflecting the coordinates, we’ll stop looking at those equations.
Instead, we’ll focus only on the “fold and stamp” idea. Let’s take a triangle and reflect across a couple of different
lines to see some examples.
We’ll begin by reflecting the triangle
across the line 1 = 2 which is a horizontal line
at a height of 2 as seen to the left. Imagine
folding the paper on that red line. Where
would the blue triangle land? Stamp a green
triangle there in your mind. Did you get the
picture to the right?
Now let’s reflect that same triangle across a diagonal line of 1 V.
Fold and stamp and you should get the picture to the left. Fold and stamp!
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A Series of Transformations
We can also apply multiple transformations to a shape. When we do so, we apply only one transformation
at a time IN THE ORDER THEY ARE GIVEN. So if we wanted to dilate, translate, and then reflect we would first dilate
the pre-image, then translate that new picture, then reflect that translation. For now we’ll stick with just the two
transformations we have covered so far.
Reflect-Reflect
Taking a triangle, we can apply a reflection across the 1-axis and then reflect that across the -axis:
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6
A
A'
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A
4
B
C
2
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A'
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A
4
4
C'
B'
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B
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C
2
5
2
2
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8
Reflection across p-axis
C'
B'
C''
B''
B
C
2
5
5
2
5
C''
B''
4
A''
A''
6
8
Reflect that across n-axis
Final Picture of the Series
Notice that the in-between step is not shown on the final graph. You are welcome to leave the in-between
step as you are graphing. Just be sure to label appropriately so we know which is which.
Also notice that we now have double prime to represent the second reflection and we end up with the final
image points of ^′′: (6, −5), j′′: (4, −2), X′′: (2, −2). You might notice that this also looks like a rotation by 180°
which is true. It turns out that reflecting across both axes gives you a 180° turn.
Dilate-Reflect
Let’s again take that same triangle and dilate by 0 = with a center of dilation of (0,0) and then reflect
across the 1-axis. Keeping in mind the order, we must dilate first because that was the order listed. Then we’ll take
that dilated image and reflect it. Take a look. Shrink then fold and stamp that shrunken triangle.
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8
6
A
6
A
4
C'
B'
5
B
C2
C' C'' B''
5
C'' B''
5
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2
2
2
4
4
4
6
6
6
8
8
8
r
Dilation by q = s
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A''
C2
B
5
4
A''
A'
C2
B'
6
A
4
A'
B
8
Reflect that across p-axis
Final Picture of the Series
Lesson 2.2
Perform the given reflection or series of transformations on each given pre-image. The line of reflection is marked as a red
line. When performing a dilation, use the origin as the center of dilation.
1. Reflect across -axis
2. Reflect across = 3
3. Reflect across 1 V
4. Reflect across 1-axis
5. Reflect across 1 2
6. Reflect across 1 7. Reflect across -axis
8. Reflect across V2
9. Reflect across 1 49
10. Reflect across -axis
11. Reflect across 1-axis
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8
13. Reflect across -axis
14. Reflect across -axis
15. Reflect across 1-axis
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8
16. Reflect across 1 = −2
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17. Reflect across = −2
8
18. Reflect across 1 = 8
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4
4
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2
2
5
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12. Reflect across 1-axis
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2
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6
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8
19. Reflect across -axis
and dilate by 0 =
8
20. Reflect across 1-axis
and dilate by 0 =
21. Reflect across 1-axis
and dilate by 0 =
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22. Dilate by 0 = 2
and reflect across -axis
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23. Dilate by 0 =
and reflect across 1-axis
8
24. Dilate by 0 =
and reflect across -axis
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4
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2
2
2
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5
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2
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8
25. Reflect across -axis
and dilate by 0 =
8
26. Reflect across 1-axis
and dilate by 0 = 27. Reflect across -axis
and dilate by 0 = 8
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6
6
4
4
4
2
2
2
5
5
5
8
5
5
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2
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8
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28. Reflect the triangle across the line = 2. Which of the
following statements will be true? For each explain why or
why not.
a. Point ^′ is at the same coordinates as Point ^.
b. Point j′ is at the same coordinates as Point j.
c. Point X′ is at the same coordinates as Point X.
d. The image’s perimeter equals the pre-image’s.
e. The image’s area equals the pre-image’s.
f. The image is congruent to pre-image.
g. Line segment mmmmm
^′X′ is horizontal.
29. Reflect the triangle across the line 1 . Which of the
following statements will be true? For each explain why or
why not.
a. Point ^′ will be at the same coordinates as Point ^.
b. Point j′ will be at the same coordinates as Point j.
c. Point X′ will be at the same coordinates as Point X.
d. The image’s perimeter equals the pre-image’s.
e. The image’s area equals the pre-image’s.
f. The image is similar to pre-image.
g. Line segment mmmmmm
j′X′ is vertical.
30. In your own words, explain what a reflection does to a pre-image. Remember to consider the line of
reflection in your explanation.
31. How could reflections be used in real life?
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