Transformations on the Transformations on the coordinate planecoordinate plane

Transformations on the Transformations on the coordinate planecoordinate plane

Transformations ReviewTransformations ReviewTransformations ReviewTransformations ReviewTypeType DiagramDiagram

A A translationtranslation moves a moves a figure left, right, up, or figure left, right, up, or downdown

A A reflectionreflection moves a moves a figure across its line of figure across its line of reflection to create its reflection to create its mirror image.mirror image.

A A rotationrotation moves a moves a figure around a given figure around a given point.point.

Now we will look at how each Now we will look at how each transformation looks on a coordinate transformation looks on a coordinate

plane. The transformed figure is often plane. The transformed figure is often named with the same letters, but adding named with the same letters, but adding an apostrophe. The transformation of an apostrophe. The transformation of

ABCABC is is A’B’C’A’B’C’..

TranslationTranslationTranslationTranslation

Translate Translate ABCABC 6 units to the right. 6 units to the right.Translate Translate ABCABC 6 units to the right. 6 units to the right.

6 UnitsA

B

C

A’

B’

C’

Find point Find point AA and and

Find point Find point BB and and

Find point Find point CC and and

count 6 units to the count 6 units to the right.right. Plot point Plot point A’A’..

count 6 units to the count 6 units to the right.right. Plot point Plot point B’B’..

count 6 units to the count 6 units to the right.right. Plot point Plot point C’C’..

Translation RulesTranslation Rules•To translate a figure To translate a figure aa units to the right, units to the right, increase the increase the xx-coordinate of each point by -coordinate of each point by aa amount.amount.

•To translate a figure To translate a figure aa units to the right, units to the right, increase the increase the xx-coordinate of each point by -coordinate of each point by aa amount.amount.

Translate point P (3, 2) 9 units to the right.Since we are going to the right, we add 9 to

the x-coordinate. 3 + 9 = 12, so the new coordinates of P’ are (12, 2)

•To translate a figure To translate a figure aa units to the left, units to the left, decrease the x-coordinate of each point by a decrease the x-coordinate of each point by a amount.amount.

•To translate a figure To translate a figure aa units to the left, units to the left, decrease the x-coordinate of each point by a decrease the x-coordinate of each point by a amount.amount.

Translate point P (3, 2) 6 units to the left.Since we are going up, we subtract 6 to the x-coordinate. 3 - 6 = -3, so the new coordinates

of P’ are (-3, 2)

Translation RulesTranslation Rules•To translate a figure To translate a figure aa units up, units up, increase increase the the yy-coordinate of each point by -coordinate of each point by aa amount. amount.•To translate a figure To translate a figure aa units up, units up, increase increase the the yy-coordinate of each point by -coordinate of each point by aa amount. amount.

Translate point P (3, 2) 9 units up.Since we are going up, we add 9 to the y-

coordinate. 2 + 9 = 11, so the new coordinates of P’ are (3, 11)

•To translate a figure To translate a figure aa units down, units down, decrease decrease the the yy-coordinate of each point by a amount.-coordinate of each point by a amount.•To translate a figure To translate a figure aa units down, units down, decrease decrease the the yy-coordinate of each point by a amount.-coordinate of each point by a amount.

Translate point P (3, 2) 6 units down.Since we are going down, we subtract 6 to the

y-coordinate. 2 - 6 = -4, so the new coordinates of P’ are (3, -4)

Translation Example 2Translation Example 2

The coordinates of The coordinates of point point AA are (-5, 4) are (-5, 4)

Since we are Since we are moving to the right moving to the right we increase the we increase the xx--coordinate by 6.coordinate by 6.

The coordinates of The coordinates of point point BB are (-2, 3) are (-2, 3)

Since we are Since we are moving to the right moving to the right we increase the we increase the xx--coordinate by 6.coordinate by 6.

The coordinates of The coordinates of point point CC are (-3, 1) are (-3, 1)

Since we are Since we are moving to the right moving to the right we increase the we increase the xx--coordinate by 6.coordinate by 6.

-5 + 6 = 1-5 + 6 = 1, so the , so the new coordinates new coordinates of of A’A’ are are (1, 4)(1, 4)..

-2 + 6 = 4-2 + 6 = 4, so the , so the new coordinates new coordinates of of B’B’ are are (4, 3)(4, 3)..

-3 + 6 = 3-3 + 6 = 3, so the , so the new coordinates new coordinates of of C’C’ are are (3, 1)(3, 1)..

PracticePractice

• Point Point PP (5, 8). Translate 2 to the left and 6 (5, 8). Translate 2 to the left and 6 up.up.

• Point Z (-3, -6). Translate 5 to the right Point Z (-3, -6). Translate 5 to the right and 9 down.and 9 down.

• Translate Translate LMNLMN, whose coordinates are , whose coordinates are (3, 6), (5, 9), and (7, 12), 9 units left and (3, 6), (5, 9), and (7, 12), 9 units left and 14 units up.14 units up.

P’P’ (3, 14) (3, 14)

Z’Z’ (2, -15) (2, -15)

L’M’N’L’M’N’ (-6, 20), (-4, 23), (-2, 26) (-6, 20), (-4, 23), (-2, 26)

ReflectionReflectionReflectionReflection

Reflect Reflect ABCABC across the across the yy-axis.-axis.Reflect Reflect ABCABC across the across the yy-axis.-axis.

5 Units

2 UnitsA

B

C

5 Units A’

2 Units

B’

3 Units 3 UnitsC’

Count the number of units point Count the number of units point AA is from the line of reflection.is from the line of reflection.Count the same number of units on Count the same number of units on the other side and plot point the other side and plot point A’A’..

Count the number of units point Count the number of units point BB is from the line of reflection.is from the line of reflection.Count the same number of units on Count the same number of units on the other side and plot point the other side and plot point B’B’..

Count the number of units point Count the number of units point CC is from the line of reflection.is from the line of reflection.Count the same number of units on Count the same number of units on the other side and plot point the other side and plot point C’C’..

Reflection RulesReflection Rules•To reflect point To reflect point (a(a, , bb) across the ) across the yy-axis use -axis use the opposite of the the opposite of the xx-coordinate and keep -coordinate and keep the the y y coordinate the same.coordinate the same.

•To reflect point To reflect point (a(a, , bb) across the ) across the yy-axis use -axis use the opposite of the the opposite of the xx-coordinate and keep -coordinate and keep the the y y coordinate the same.coordinate the same.

Reflect point P (3, 2) across the y-axis.Since we reflecting across the y-axis. Keep

the y the same and use the opposite of the x. (-3, 2)

•To reflect point To reflect point (a(a, , bb) across the ) across the xx-axis -axis keep the keep the x-x-coordinate the same and use the coordinate the same and use the opposite of the opposite of the yy-coordinate-coordinate

•To reflect point To reflect point (a(a, , bb) across the ) across the xx-axis -axis keep the keep the x-x-coordinate the same and use the coordinate the same and use the opposite of the opposite of the yy-coordinate-coordinate

Reflect point P (3, 2) across the x-axis.Since we reflecting across the x-axis. Keep

the x the same and use the opposite of the y. (3, -2)

PracticePracticeThe coordinates of The coordinates of ABCABC are: are:

(-5, 4), (-2, 3), (-3, 1) (-5, 4), (-2, 3), (-3, 1) Reflect Reflect ABC ABC across the across the yy-axis and -axis and

then reflect it across the then reflect it across the xx-axis.-axis.To reflect it across the To reflect it across the yy-axis keep the y the same -axis keep the y the same and use the opposite x. The new coordinates are: and use the opposite x. The new coordinates are:

(5, 4), (2, 3), (3, 1)(5, 4), (2, 3), (3, 1)To reflect it across the To reflect it across the xx-axis keep the x the same -axis keep the x the same and use the opposite y. The new coordinates are: and use the opposite y. The new coordinates are:

(5, -4), (2, -3), (3, -1)(5, -4), (2, -3), (3, -1)

Rotation RulesRotation Rules•To rotate a point 90° clockwise, To rotate a point 90° clockwise, switch the switch the coordinates, and then multiply the new coordinates, and then multiply the new yy--coordinate by -1.coordinate by -1.

•To rotate a point 90° clockwise, To rotate a point 90° clockwise, switch the switch the coordinates, and then multiply the new coordinates, and then multiply the new yy--coordinate by -1.coordinate by -1.Rotate point P (3, 2) clockwise about the origin.Since we are rotating it clockwise, we switch the coordinates (2, 3) and multiply the new y

by -1, so the new coordinates are (2, -3)

•To rotate a point 180°, To rotate a point 180°, just multiply each just multiply each coordinate by -1.coordinate by -1.•To rotate a point 180°, To rotate a point 180°, just multiply each just multiply each coordinate by -1.coordinate by -1.Rotate point P (3, 2) clockwise about the origin.

Since we are rotating it 180°, we simply multiply the coordinates by -1, so the new

coordinates are (-3, -2).

PracticePractice

• Point Point PP (5, 8). Rotate 90° clockwise about (5, 8). Rotate 90° clockwise about the origin.the origin.

• Point Z (-3, -6). Rotate 180° about the Point Z (-3, -6). Rotate 180° about the origin.origin.

• Rotate Rotate LMNLMN, whose coordinates are (3, , whose coordinates are (3, 6), (5, 9), and (7, 12), 90° clockwise about 6), (5, 9), and (7, 12), 90° clockwise about the origin.the origin.

P’P’ (8, -5) (8, -5)

Z’Z’ (-3, -6) (-3, -6)

L’M’N’L’M’N’ (6, -3), (9, -5), (7, -12) (6, -3), (9, -5), (7, -12)