9-2 translations you found the magnitude and direction of vectors. draw translations. draw...
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9-2 Translations
You found the magnitude and direction of vectors.
• Draw translations.
• Draw translations in the coordinate plane.
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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.
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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).
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Direction and Length
From the school entrance, I went three blocks north.
The distance (magnitude) is:
Three blocksThe direction is:North
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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).
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B
A45°
60°
N
S
EWA
B
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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
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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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p. 632
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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:
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Which of the following shows the translation of ΔABC along the translation vector?
A. B.
C. D.
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p. 633
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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:
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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:
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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.
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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:
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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
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9-2 Assignment
Page 627, 10-14 even, 20, 21, 26, 27