SECTION 9.2Translations

In Lesson 4.7, you learned that a translation or slide is a transformation that moves all points of a figure the same distance in the same direction. Since vectors can be used to describe both distance and direction, vectors can be used to define translations.

Example 1: Draw the translation of the figure along the translation vector.

Step 1 Draw a line through eachvertex parallel to vector .w

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Step 2 Measure the length ofvector . Locate point G'by marking off this distancealong the line throughvertex G, starting at G andin the same direction as thevector.

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Step 3 Repeat Step 2 to locate points H', I', and J' to form the translated image.

Recall that a vector in the coordinate plane can be written as , where a represents the horizontal change and b is the vertical change from the vector’s tip to its tail. is represented by the ordered pair .Written in this form, called the component form, a vector can be used to translate a figure in the coordinate plane.

,a b

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2, 4

Example 2:

a) Graph ΔTUV with vertices T(–1, –4), U(6, 2), and V(5, –5) along the vector –3, 2.

The vector indicates a translation 3 units left and 2 units up.

(x, y) → (x – 3, y + 2)

T(–1, –4) → (–4, –2)

U(6, 2) → (3, 4)

V(5, –5) → (2, –3)

Example 2:

b) Graph pentagon PENTA with vertices P(1, 0), E(2, 2), N(4, 1), T(4, –1), and A(2, –2) along the vector –5, –1.

The vector indicates a translation 5 units left and 1 unit down.

(x, y) → (x – 5, y – 1)

P(1, 0) → (–4, –1)

E(2, 2) → (–3, 1)

N(4, 1) → (–1, 0)

T(4, –1) → (–1, –2)

A(2, –2) → (–3, –3)

Example 3: The graph shows repeated translations that result in the animation of the raindrop.

a) Describe the translation of the raindrop from position 2 to position 3 in function notation and in words.

The raindrop in position 2 is (1, 2). In position 3, this point moves to (–1, –1). Use the translation function (x, y) → (x + a, y + b) to write and solve equations to find a and b.

(1 + a, 2 + b) or (–1, –1)

1 + a = –1 2 + b = –1

a = –2 b = –3

Answer: function notation: (x, y) → (x – 2, y – 3). So, the raindrop is translated 2 units left and 3 units down from position 2 to 3.

Example 3: The graph shows repeated translations that result in the animation of the raindrop.

b) Describe the translation of the raindrop from position 3 to position 4 using a translation vector.

(–1 + a, –1 + b) or (–1, –4)

–1 + a = –1 –1 + b = –4

a = 0 b = –3

Answer: translation vector: 0, 3