9-2 Translations

You found the magnitude and direction of vectors.

• Draw translations.

• Draw translations in the coordinate plane.

DefinitionA translation is a transformation that

moves all the points in a plane a fixed distance in a given direction (slide).

The arrow shows the direction of the translation.

Definition•

A

B

Initial point or tail

Terminal point or tip

A vector can be represented as a “directed” line segment, useful in describing paths.

A vector has both direction and magnitude (length).

Direction and Length

From the school entrance, I went three blocks north.

The distance (magnitude) is:

Three blocksThe direction is:North

Direction and Magnitude

The magnitude of AB is the distance between A and B.

The direction of a vector is measured counterclockwise from the horizonal (positive x-axis).

B

A45°

60°

N

S

EWA

B

Drawing Vectors

Draw vector YZ with direction of 45° and length of 10 cm.

1.Draw a horizontal dotted line2.Use a protractor to draw 45° 3.Use a ruler to draw 10 cm4.Label the points

45°

Y

Z

10 c

m

Translation vectorSince vectors have a distance and a direction, they

are often used to describe translations. The vector shows the direction of the translation

and its length gives the distance each point travels.

To measure direction, add a horizontal dotted line and measure counterclockwise

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Draw a TranslationCopy the figure and given translation vector. Then draw the translation of the figure along the translation vector.

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.

Step 1 Draw a line through eachvertex parallel to vector .

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

Answer:

Which of the following shows the translation of ΔABC along the translation vector?

A. B.

C. D.

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Translations in the Coordinate PlaneA. 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)

Answer:

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)

Answer:

Describing TranslationsA. ANIMATION The graph shows repeated translations that result in the animation of the raindrop. 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.

B. ANIMATION The graph shows repeated translations that result in the animation of the raindrop. 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:

B. The graph shows repeated translations that result in the animation of the soccer ball. Describe the translation of the soccer ball from position 3 to position 4 using a translation vector.

A. –2, –2

B. –2, 2

C. 2, –2

D. 2, 2

9-2 Assignment

Page 627, 10-14 even, 20, 21, 26, 27