1. Topic 2 Summary
Transformations

2. Reflections You may be asked to reflect about: 1) x axis 2) y axis 3) line y=x 4) line y=-x 5) any horizontal line such as y=3 6) any vertical line such as x=2

3. Reflecting across x-axis
Start by reflecting one point, lets say point M, since M is 1 unit away from the x axis, its reflection will also be 1 unit away from the x axis. Notice the pattern of how x and y change and apply this pattern to the other vertices.
Reflecting across x-axis
Pre-image
Image T H
M(2, 1)
T(-3, 5) A M
T’(-3, -5)
M’(2, -1) A’ M’ T’ H’
A(-1, 1)
H(4, 5)
A’(-1,-1)
H’(4, -5)
If correct, the line of reflection which in this example is the x axis should be right in the middle of the two shapes.

4. Sometimes you will have to reflect about horizontal or vertical lines other than the x or y axis.
Remember that if the equation is y= then x could be any number as long as you keep y constant.
y = 2 is horizontal line because as you look at points on that line, the value of x changes but the y is always 2
y=2

5. Remember that if the equation is x= then y could be any number as long as you keep x constant.
x = 2 is vertical line because as you look at points up or down this line, the y values change but the x is always 2.
x=2

6. Reflecting across a vertical line
Reflect across x = 2
1) Draw line of reflection A B B'
2) Pick a starting point,
count how far it is from line of reflection A' D C C' D'
3) Go that same distance
on the other side of line
4) Label the new point
5) Continue with other points

7. Reflect across y = -3
After drawing the line, pick a starting point, lets say A and count how far it is from line of reflection, in this case point A is 4 units from the line of reflection. Then count 4 units on the other side of the line to locate the reflection A’ H T A
A’(7, -7)
H’(-12, 2)
T’(2, -7)

8. Memorize the formula (x,y) -> (y,x)
Reflecting across the line y = x
Memorize the formula
(x,y) -> (y,x)
Pre-Image
Image
F(-3, 0)
F‘(0, -3) I’ S’
I(4, 0) F I
I'(0, 4)
S(4, -9) F’ H’
S'(-9, 4)
H(-3, -9) H S
H'(-9, -3)

9. Reflect across y = –x Memorize the formula (x,y) -> (-y,-x) E’ E V
V(0, 6) E(-7, 6)
V’(-6, 0) E’(-6, -7) M O O’ V’

10. Rotations
Remember that each time a point is rotated 90 degrees, two things happen:
The point moves to the quadrant to the left or to the right depending on the direction of the rotation (clockwise vs counterclockwise)
The values of x and y of the original point (preimage) switch places.

11. Example of a rotation Rotate point P (-3,2) counterclockwise 90 degrees.
1) Point P is on the second quadrant and rotating 90 degrees CCW will move it to the third where both the X and the Y coordinates are negative. 2) Then switch the x and the y to obtain P’ at (-2,-3).

12. Translations
When moving to the right or to the left the x value changes. To the right you add, to the left, you would subract.
When moving up or down, the y value of the point changes. Up the y value increases, so you add. Down the y value decreases, so you would subtract.
Examples:
The rule for Left 2, Up 3 would be (x-2,y+3)
The rule for Right 1, Down 10 would be (x+1,y-10)