Transformations Unit 7

Section 1 Introduction to Transformations & Translations

What is happening in these pictures? Insert picture of translation and rotation here

Do Now Take out a sheet of paper and answer the following questions: What areas did I do well on in the test? What areas do I still need to work on? What can I do to improve/keep up the good work for the next test?

What is happening in these pictures? Insert picture of reflection and dilation here

Transformations Transformation—the process of moving a figure or changing it Map—to move The location of the figure after it is transformed is called the transformation image Example: The image of point A after it is mapped through a translation is point A’.

Four Main Types of Transformations Translation Rotation Reflection Dilation

Translation Animation 1 What is happening in the animation?

Definition of Translation Moving a figure to a new location with no other changes Remember: Translate = Slide Every point moves the same number of units in the same direction

Notation Transformations: add apostrophe to letter for each point: Original Object (A)  Transformation Image (A’) Example: Translation of triangle ABC is triangle A’B’C’

Translations on the Coordinate Plane Continued (x + h, y + k) Positive = shift up or to the right Negative = shift down or to the left Also can be written as <h, k> Ex.: Point A(-1, -5) transforms according to the rule (x, y)  (x + 2, y + 3). What is the resulting point? Answer: A’(-1 + 2, -5 + 3) = A’(1, -2)—two units right, three units up

Practice Problems I (-6, 6) (-2, -11) State the transformation image (result) after each transformation. Describe what is happening in words. (-1, 3)  (x – 5, y + 3)  (5, -12)  (x – 7, y + 1)  (-6, 6) (-2, -11)

Practice Problems II A triangle with coordinates L(-3, 2), M(6, -2), and N(2, -5) is translated so that M’ is at (2, 2). Where are the images of points L and N located? From M to M’ is (x – 4, y + 4) L’(-3 – 4, 2 + 4) = L’(-7, 6) N’(2 – 4, -5 + 4) = N’(-2, -1)

Section 2 Rotations

How are these motions different?

Rotation A rotation is a transformation where the figure is turned about a given point. The center of rotation is the point around which the object is rotated.

Rotation in the Coordinate Plane Instead of left/right, we say clockwise (cw) or counterclockwise (ccw) Animation 2 A clockwise rotation of 90 degrees is equal to a counterclockwise rotation of how many degrees? Answer: 270 degrees

How to Rotate a Shape a Graph Question: Graph the rotation of triangle ABC with points A(2, 1); B(5, 2); C(3, 4) 90 degrees ccw. Plot the axes and points on your graph. Rotate your paper 90° clockwise. If the question says cw, rotate your paper ccw. Plot the same coordinates, using the y-axis as the x-axis and the x-axis as the y-axis. What do you notice about the coordinates?

Practice Activity Graph the image of triangle ABC with points A(2, 1); B(5, 2); C(3, 4) under a rotation of… 180° ccw 270° ccw What do you notice about the coordinates?

Special Rotations 90° ccw (270° cw)—image has coordinates (-y, x); (2, 1)  (-1, 2) 180° ccw or cw—image has coordinates (-x, -y); (2, 1)  (-2, -1) 270° ccw (90° cw)—image has coordinates (y, -x); (2, 1)  (1, -2)

Practice Problems Find the coordinates of these points when rotated 90° ccw, 180° ccw, and 270° ccw : Point 90° ccw 180° ccw 270° ccw (-3, -5) (2, 1) (0, -4) (5, -3) (3, 5) (-5, 3) (-1, 2) (-2, -1) (1, -2) (4, 0) (0, 4) (-4, 0)

Section 3 Reflections

Reflection A transformation where each point appears at an equal distance on the opposite side of a given line Line of reflection Animation 1

What do you notice? What parts of the shape stay the same? Which part changes? Orientation is reversed—the object is no longer facing the same way

Graphing a Reflection on the Coordinate Plane Question: Graph the reflection of triangle ABC with points A(-2, 1); B(-5, 2); and C(-3, 4) across the x-axis. Graph the axes and plot the points. Fold your paper along the line of reflection. Trace the triangle on the back of your paper. Unfold the paper and trace the triangle again, this time on the front of the paper. Label the new coordinates.

On Your Own… Graph the reflection of triangle ABC with points A(-2, 1); B(-5, 2); and C(-3, 4): Across the y-axis Across the line y = x (Hint: Graph the line first!)

What are the coordinates of the transformation image? Find the coordinates of the image after a reflection across x-axis, y-axis, and line y = x. Point x-axis y-axis y = x A(-2, 1) B(-5, 2) C(-3, 4) A’(-2, -1) B’(-5, -2) C’(-3, -4) A’’(2, 1) B’’(5, 2) C’’(3, 4) A’’’(1, -2) B’’’(2, -5) C’’’(4, -3)

Special Reflections Across the x-axis: (x, y)  (x, -y) Example: (-2, 1)  (-2, -1) Across the y-axis: (x, y)  (-x, y) Example: (-2, 1)  (2, 1) Across the line y = x: (x, y)  (y, x) Example: (-2, 1)  (1, -2)

Section 4 Dilations

What is happening in this picture?

Dilation Transformation in which a polygon is enlarged or reduced by a factor around a center point. “Zoom in” “Zoom out”

What kind of shapes do dilations create? SIMILAR “Same shape, different sizes” Congruent angles Proportional sides

Isometry Transformation that results in a congruent image Examples of isometry: Translation Rotation Reflection NOT an example of isometry Dilation

Different Kinds of Isometry Direct isometry—orientation stays the same Translation Rotation Opposite isometry—orientation is reversed Reflection

Dilation in the Coordinate Plane To find the dilation image, multiply every point by the scale factor Scale Factor—how big the image is compared to the original AKA. Dilation ratio Example: Point A(-3, 2) undergoes a dilation of 2. What are the coordinate of the image? Answer: (-3 × 2, 2 × 2) = (-6, 4)

Practice Problems Original Scale factor = ½ Scale factor = 2 D’(½, 1) A’(-1.5, -2.5) A’’(-6, -10) A’’’(-9, -15) B(-7, 5) B’(-3.5, 2.5) B’’’(-21, 15) C(0, -4) C’(0, -2) C’’(0, -8) D(1, 2) D’’(2, 4) D’’’(3, 6)

Section 5 Compositions

Composition Combination of two or more transformations Examples: Dilation and translation Reflection and rotation

What happens when we combine two or more transformations? What is happening in this picture? Glide reflection = reflection + translation

Notation D2 – Dilation of 2 R90 – Rotation of 90° counterclockwise Assume the direction is ccw unless it says “clockwise” ry-axis – Reflect across the y-axis T-2,3 – Translation of (x – 2, y + 3)

How to Do Composition Problems Treat composition as two problems DO SECOND PART FIRST! Example: A point located at A(-3, -2) undergoes the following composition: D2◦ R90. State the image of points A’. R90: Use formula (-y, x). A(-3, -2)  A’(2, 3) D2 : A(4, 6

Does order matter in a composition? Try doing these two compositions. What do you notice about the answer? Point A(-2, -4) dilated D2 ◦ T-2,3 . A’(-8, -2) Point A (-2, -4) dilated T-2,3 ◦ D2. A’(-6, -5) ORDER MATTERS IN DILATION!

Practice Problems Find the image after each composition. D2 ◦ T-2,3 Original D2 ◦ T-2,3 R90 ◦ ry-axis ry=x◦R180 A(-2, 1)

Section 6 Coordinate Proofs with Transformations

Comparing Different Transformations Which properties are preserved under each transformation? Distance Orientation Angle Measure Parallelism Translation Rotation Reflection Dilation Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes No Yes Yes Yes

Parallelism “Is parallelism preserved?” Means: “Are parallel lines in an object still parallel after the object is transformed?” Slopes of parallel lines are equal. Use the slope formula to prove this:

Example 1 Given: quadrilateral ABCD with vertices A(-1,1), B(4,-2), C(3,-5), and D(-2,-2). State the coordinates of A'B'C'D', the image of quadrilateral ABCD under a dilation of factor 2. Prove that A'B'C'D' is a parallelogram. Dilation of 2: A’(-2, 2); B’(8, -4); C’(6, -10); D’(-4, -4)

Example 1 Continued Prove that A'B'C'D' is a parallelogram. Opposite sides of a parallelogram are parallel. Find the slopes of A’B’ and D’C’ to show that they’re equal. Find the slopes of B’C’ and A’D’ to show that

Example 2 The vertices of triangle ABC are A(2,1), B(7,2), and C(3,5). Identify and graph a transformation of triangle ABC such that its image results in AB||A’B’. C B A