warm up 1 1. graph Δabc with the vertices a(–3, –2), b(4, 4), c(3, –3) 2. graph Δabc with...
TRANSCRIPT
WARM UP
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1. Graph ΔABC with the vertices A(–3, –2), B(4, 4), C(3, –3)
2. Graph ΔABC with the vertices D(1, 2), E(8, 8), F(7, 1)
Compare the two graphs and write a few sentences describing the similarities and differences of the two triangles.
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Unit 2-Lesson 1
Unit 2:Transformations
Lesson 1: Reflections and Translations
Objectives
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• I can identify and perform reflections and translations on a coordinate plane.
• I can predict the effect of a given ridged motion on a given figure.
• I can determine if two figures are congruent after a congruence transformation.
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Types of Transformations
Reflections: These are like mirror images as seen across a line or a point.
Translations ( or slides): This moves the figure to a new location with no change to the looks of the figure.
Rotations: This turns the figure clockwise or counter-clockwise but doesn’t change the figure.
Dilations: This reduces or enlarges the figure to a similar figure.
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Reflections
You could fold the picture along line l and the left figure would coincide with the corresponding parts of right figure.
l
You can reflect a figure using a line or a point. All measures (lines and angles) are preserved but in a mirror image.
Example: The figure is reflected across line l .
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Reflections – coordinates…
reflects across the y axis to line n
(2, 1) (-2, 1) & (5, 4) (-5, 4)
Reflection across the x-axis: the x values stay the same and the y values change sign. (x , y) (x, -y)
Reflection across the y-axis: the y values stay the same and the x values change sign. (x , y) (-x, y)
Example: In this figure, line l :
reflects across the x axis to line m.
(2, 1) (2, -1) & (5, 4) (5, -4)
ln
m
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Reflections across specific lines:
To reflect a figure across the line y = a or x = a, mark the corresponding points equidistant from the line.
i.e. If a point is 2 units above the line its corresponding image point must be 2 points below the line.
B(-3, 6) B′ (-3, -4)
C(-6, 2) C′ (-6, 0)
A(2, 3) A′ (2, -1).
Example:
Reflect the fig. across the line y = 1.
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Lines of Symmetry If a line can be drawn through a figure so the one side of the
figure is a reflection of the other side, the line is called a “line of symmetry.”
Some figures have 1 or more lines of symmetry. Some have no lines of symmetry.
One line of symmetry
Infinite lines of symmetry
Four lines of symmetry
Two lines of symmetry
No lines of symmetry
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• translation vector – shows direction and distance of the “slide”
VECTOR INTRODUCTION
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Copy the figure and given translation vector. Then draw the translation of the figure along the translation vector.
Step 1 Draw a line through each vertex parallel to 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 3 Repeat Step 2 to locate points H', I', and J' to form the translated image.
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Translations (slides) If a figure is simply moved to another location without change to its
shape or direction, it is called a translation (or slide). A vector tells you how to translate a point <a, b> or <-a, -b>. If a point is moved “a” units to the right and “b” units up, then the
translated point will be at (x + a, y + b). If a point is moved “a” units to the left and “b” units down, then the
translated point will be at (x - a, y - b).
A
A′
Image A translates to image A′ by moving to the right 3 units and down 8 units.
Example:
A (2, 5) B (2+3, 5-8) A′ (5, -3)
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Translations in the Coordinate Plane
A. Graph ΔTUV with vertices T(–1, –4), U(6, 2), and V(5, –5) along the vector –3, 2.
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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)
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Translations in the Coordinate PlaneB. Graph pentagon PENTA with vertices P(1, 0), E(2, 2), T(4, –1), and A(2, –2) along the vector–5, –1.
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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)
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A. A'(–2, –5), B'(5, 1), C'(4, –6)
B. A'(–4, –2), B'(3, 4), C'(2, –3)
C. A'(3, 1), B'(–4, 7), C'(1, 0)
D. A'(–4, 1), B'(3, 7), C'(2, 0)
A. Graph ΔABC with the vertices A(–3, –2), B(4, 4), C(3, –3) along the vector –1, 3. Choose the correct coordinates for ΔA'B'C'.
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B. Graph □GHJK with the vertices G(–4, –2), H(–4, 3), J(1, 3), K(1, –2) along the vector 2, –2. Choose the correct coordinates for □G'H'J'K'.
A. G'(–6, –4), H'(–6, 1), J'(1, 1), K'(1, –4)
B. G'(–2, –4), H'(–2, 1), J'(3, 1), K'(3, –4)
C. G'(–2, 0), H'(–2, 5), J'(3, 5), K'(3, 0)
D. G'(–8, 4), H'(–8, –6), J'(2, –6), K'(2, 4)
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A. 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.