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Transformations in the Coordinate Plane

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                            Transformations in the
                            Coordinate Plane
                            BEFORE                              Now                                   WHY?

                            You identified translations,        You’ll graph transformations          So you can describe movement
                            reflections, and rotations.         in a coordinate plane.                in a cartoon, as in Ex. 17.


                                     In the Real World

   Word Watch                            Computer Graphics Computer
Review Words                             and video game programmers use                                 y
                                         transformations to create patterns                         6
coordinate plane, p. 293                                                                            5
transformation, p. 511                   and animations. How can you use
                                                                                                    4
image, p. 511                            coordinates to describe the
                                                                                                    3
translation, p. 511                      transformation shown?                                      2
reflection, p. 511
                                                                                                    1
rotation, p. 511                         The transformation is a translation.
                                         You can use coordinate notation                           O        1 2 3 4 5 6 7 8 x
                                         to describe a translation. An arrow
                                         is used in coordinate notation to
                                         signify “goes to.”

                                                                    Coordinates of           Coordinates
                                                                    original point            of image

                                                                        (x, y) → (x            a, y          b)

                                                 The number a tells how many units             The number b tells how many units
                                                 to shift the figure to the left or right.     to shift the figure up or down.




                                     EXAMPLE          1       Describing a Translation

                                         You can use coordinate notation to describe the translation shown
                                         above. Each point on the original figure is moved 4 units to the left
                                         and 3 units down.

                                         ANSWER In coordinate notation you write this translation as:
                                            (x, y) → (x       ( 4), y      ( 3)) or (x, y) → (x             4, y   3)



                                     Your turn now            Describe the translation using coordinate notation.
                                           1. A figure is moved 5 units to the right and 4 units up.
                                           2. A figure is moved 7 units to the left and 2 units up.



518      Chapter 10        Geometric Figures
                                                                                                                                       Page 2 of 5




                                EXAMPLE         2         Translating a Figure

     with   Reading               Draw quadrilateral JKLM with vertices J( 5, 3), K( 4, 5),
Each point on an image is         L( 3, 3), and M( 4, 1). Then find the coordinates of
labeled with a prime. The         the vertices of the image after the translation
notation J is read “J prime.”     (x, y) → (x 6, y 2), and draw the image.
                                  For each vertex of the original figure, add 6 to the x-coordinate
                                  and subtract 2 from the y-coordinate.
                                  Original           Image                                                                         y
                                                                                                                           6
                                  J( 5, 3) → J (1, 1)                                                  K                   5

                                  K( 4, 5) → K (2, 3)                                          J               L
                                                                                                                           4
                                                                                                                                           K
                                                                                                                           3
                                  L( 3, 3) → L (3, 1)                                                                      2
                                                                                                                           1
                                                                                                                                   J                 L
                                  M( 4, 1) → M (2,                1)                                   M
                                                                                           6       5   4   3       2       O           1         3       4 x
                                  Each point on the original                                                                               M
                                  figure is translated 6 units                                                             2
                                  to the right and 2 units down.
                                  The graph shows both figures.



                                Your turn now             Complete the following exercise.
                                   3. Draw       ABC with vertices A( 4, 0), B(0, 4), and C(0, 0). Then
                                        find the coordinates of the vertices of the image after the
                                        translation (x, y) → (x 4, y 6), and draw the image.



                                  Reflections You can also use coordinate notation to describe
                                  reflections.

                                                          y                                                                y
                                                      3                                                                3
                                        W( 3, 2)                                               E ( 3, 2)                           E(3, 2)
                                                      2                                                                2
                                                                  X(2, 1)
                                                      1                                                                1

                                    5   4   3   2     O       1   2    3 x                 5    4      3               O       1           3 x

                                                      2       X (2,     1)
                                   W ( 3,       2)                                         F ( 1,          2)                  F(1,         2)
                                                      3                                                                3


                                  Reflection in the x-axis:                             Reflection in the y-axis:
                                  Multiply y-coordinate by                   1.         Multiply x-coordinate by                                         1.
                                   (x, y) → (x, 1 p y) or                                (x, y) → ( 1 p x, y) or
                                   (x, y) → (x, y)                                       (x, y) → ( x, y)



                                                              Lesson 10.8         Transformations in the Coordinate Plane                                      519
                                                                                                                                     Page 3 of 5




                            EXAMPLE                 3         Reflecting a Figure

                               Draw parallelogram ABCD with vertices A( 3, 3), B(2, 3), C(4, 1),
                               and D( 1, 1). Then find the coordinates of the vertices of the
                               image after a reflection in the x-axis, and draw the image.
                               For each vertex of the original figure, multiply the y-coordinate by                                                1.
                               Original               Image                              A                   y           B
                               A( 3, 3) → A ( 3,                     3)                                  2

                               B(2, 3)              → B (2,         3)                               D                           C
                                                                                    4        3           O       1   2       3           x
                               C(4, 1) → C (4,                      1)                               D                           C
                                                                                                         2
                               D( 1, 1) → D ( 1,                         1)
                                                                                     A                                   B
                               The graph shows both figures.




                                                                                                                             INTERNET
      Exercises                                                                                                              eWorkbook Plus
                                                                                                                             CLASSZONE.COM
      More Practice, p. 714


                          Getting Ready to Practice
                              1. Vocabulary Copy and complete: The figure that results from
                                                                                                    ?
                                   performing a transformation on an original figure is called the __.

                            Describe the translation using coordinate notation.
                              2.                         y                          3.                           y
                                                M    4
                                                                                                             B
                                                                                                                                     B
                                                     3                                                       2
                                    L                           M                            A               1
                                                         N                                                               A
                                                     1
                                            K            L                N                      3           O       1               4 5 x
                                        3       2    O         2          4   5 x                            C
                                                                K                                                                    C



                            Draw GHJ with vertices G ( 4, 1), H( 2, 4), and J (0, 1). Then
                            find the coordinates of the vertices of the image after the specified
                            transformation, and draw the image.
                              4. (x, y) → (x                 4, y        4)         5. (x, y) → (x                   2, y        3)
                              6. Reflect              GHJ in the x-axis.            7. Reflect               GHJ in the y-axis.


520   Chapter 10   Geometric Figures
                                                                                                                                                        Page 4 of 5




                                  Practice and Problem Solving
         with   Homework           Describe the transformation using coordinate notation.
    Example   Exercises             8.                                                                         9.                               y
                                                   M                y        M                                                                      B
       1      8–9, 17                                          4                                                                            5

       2      10–16, 18                                        3
                                                                                                                                                B
                                                               2                                                                                          C
       3      13–16                                                                                                                A        2
                                                           N N
          Online Resources                 4                   O         1         4 x                                     A                        C
          CLASSZONE.COM
                                                                                                                       4               D            1    2    3       4 x
         • More Examples                                       2
         • eTutorial Plus                                      3                                                                   D        2
                                          L                                            L


                                   Draw the triangle with the given vertices. Then find the coordinates
                                   of the vertices of the image after the specified translation, and draw
                                   the image.
                                   10. P(1, 1), Q(3, 5), R(5, 4); (x, y) → (x                                   2, y       4)
                   Art
                                   11. F( 2, 3), G(3, 3), H(3,                         1); (x, y) → (x                 3, y            6)
                                   12. L( 6, 0), M( 6,                       4), N( 3,                 4); (x, y) → (x, y                  5)

                                   Draw rectangle FGHJ with vertices F( 2, 3), G(3, 3), H(3, 1), and
                                   J ( 2, 1). Then find the coordinates of the vertices of the image
                                   after the specified transformation, and draw the image.
                                   13. (x, y) → (x                 3, y       6)                              14. (x, y) → (x                   7, y)
                                   15. Reflect FGHJ in the x-axis.                                            16. Reflect FGHJ in the y-axis.

                                   17. Motion Transformations can be used to show motion. Use coordinate
                                         notation to describe the transformation of the drummer from one
                                         picture to the next picture.
                                                   y                                       y                                   y                                      y
                                               3                                                                                                                  3
                                               2                                                                                                                  2
                                               1                                                                                                                  1

                                               O       1   2       3 x             O           1   2    3 x            O           1   2    3 x               O           1   2   3 x
s    Motion
     Photography can show          18. Explain Draw a rectangle with vertices K(0, 0), L( 3, 0), M( 3, 4), and
     motion by using repeated            N(0, 4). Find the length, width, and area of the rectangle. Then translate
     flashes to record                   the rectangle 2 units to the right. Find the length, width, and area of the
     overlapping images.
     Capturing a drum stroke             image. Explain your results.
     may require 200 flashes
     in 4 seconds. What is this    19. Critical Thinking Draw one triangle with vertices (2, 1), (3, 4), and
     rate as a unit rate?                (4, 1) and a second triangle with vertices ( 4, 1), ( 3, 4), and ( 2, 1).
                                         Describe two different transformations that could move the first
                                         triangle onto the second.

                                                                         Lesson 10.8               Transformations in the Coordinate Plane                                        521
                                                                                                                  Page 5 of 5




                                     Rotations In Exercises 20–23, the vertices of a triangle are given. Find
                                     the coordinates of the vertices of the image after a rotation of 90
                                     clockwise about the origin. The coordinate notation (x, y) → (y, x)
                                     describes such a rotation.
                                     20. F(0, 2), G( 3, 1), H( 1, 1)        21. L(2, 2), M(4,         1), N(2,            2)
                                     22. R( 3, 3), S( 3, 0),T( 1, 0)        23. W(0, 0), X(5,         3), Y(3, 4)

                                     24. Critical Thinking The vertices of quadrilateral ABCD are A( 3, 4),
                                          B(2, 4), C(3, 2), and D( 4, 1). After a translation, the coordinates
                                          of A are ( 5, 1). Describe the translation using coordinate notation.
                                          Then find the coordinates of B , C , and D .
                                     25. Challenge Line segment EF has endpoints E(4, 3) and F(4,   3). Its
                                          image after a 180 rotation has endpoints E ( 4, 3) and F ( 4, 3).
                                          Describe a 180 rotation using coordinate notation.


                                   Mixed Review
                                     Solve the equation. Check your solution. (Lesson 7.5)
                                     26. 3x     5   23         27. 8   4x     20          28. 35          3       16x
                                     29. Choose a Strategy Use a strategy from
                                                                                          Problem Solving Strategies
                                          the list to solve the following problem.
                                          Explain your choice of strategy.                s Guess, Check, and Revise
                                                                                          s Write an Equation
                                          You paid three times as much for lunch          s Look for a Pattern


                                          as you paid for a snack. Altogether, you
                                          spent $5. How much did you pay for
                                          the snack? for lunch?

                                     Basic Skills Copy and complete the statement.
                                     30. 0.53 kg     ?
                                                    __ g       31. 8.9 m    __ cm
                                                                             ?            32. 62 mL                ?
                                                                                                                  __ L


                                   Test-Taking Practice
      INTERNET                       33. Extended Response Describe a                                     y
      State Test Practice                                                                             4
                                          combination of two transformations
      CLASSZONE.COM
                                          that would move figure A to figure B.
                                                                                                      3               A
                                                                                                      2
                                          Then describe a combination of two                          1
                                          transformations that would move
                                          figure B to figure A.                       4   3       2   O       1   2       3   4 x

                                                                                              B




522         Chapter 10      Geometric Figures

				
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