Topic 2 Summary Transformations

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

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.

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

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

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

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)

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)

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’

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.

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).

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)