Geometric Transformations

What is the pre image of a Q' (5,2) if the rule that made the image is R₀, ₋₉₀ (x, y)?

We have to undo the rotation by going in the opposite direction of the given rule. So instead of using (x, y) ↦ (y, -x), we use (x, y) ↦ (-y, x). Make y coordinate negative, then switch the coordinates. To get Q (-2, 5).

How do you find the pre image of the figure if the rule that made the image is T₅,₄?

We need to work backwards. To do this, we need to do the opposite of the rule. So take the x coordinate minus 5 and the y coordinate minus 4.

Reflection Rules

over x-axis: (x, y) ↦ (x, -y); over y-axis: (x, y) ↦ (-x, y); over line y = x: (x, y) ↦ (y, x); over line y = -x: (x, y) ↦ (-y, -x)

1) regular polygon: 2) regular polygons with n sides have _-_____ reflectional symmetry/ lines of symmetry

1) a polygon that is both equiangular and equilateral; 2) n-fold

1) Definition and attributes of reflections 2) What properties do and don't they preserve?

1) def: In a reflection, all points in pre image reflected over line of reflection to make image. attributes: Image and pre image are same distance away from line of reflection. The line of reflection is perpendicular to the line segments connecting corresponding vertices of pre image and image. If a point on the pre-image lies on the line of reflection, the image of that point is the same as the pre-image/ its mapped to itself. 2) Preserve segment lengths and angle measures but not orientation or order of vertices.

1) Definition and attributes of rotations 2) What properties do and don't they preserve?

1) def: Rotations move every point of pre image through an angle of rotation about the center of rotation to make image. attributes: -When angles of rotation counterclockwise from center of rotation, angle is positive. When angles of rotation clockwise from center of rotation, angle is negative. -The distances connecting corresponding vertices to each other all pass through the center of rotation, aren't congruent, and aren't parallel. -The distance between each point on pre image and center of rotation is the same as the distance between corresponding point on image and center of rotation. -All points on figure are rotated, not just vertices. If it was rotated 360, it would map to itself. 2) They preserve angle measures and side lengths.

1) Definition and attributes of translations. 2) What properties do and don't they preserve?

1) def: Translations moves every point of pre image the same distance and direction to make image. attributes: The connecting lines of the pre image and image vertices are the same length (points moved same distance) and parallel (points moved same direction). 2) The size, shape, orientation, measurement of angles and lines are preserved.

1) reflectional symmetry / line symmetry: 2) figure is said to have n fold reflectional symmetry where n is the _______ of lines of symmetry 3) symmetry:

1) in a figure, the state of being able to be reflected across a line and thereby produce an image that maps precisely onto the original / where a figure is mapped onto itself across a line of symmetry; 2) number 3) the property of having an object flipped, rotated, or moved to exactly match itself

1) rotational symmetry: ; 2) regular polygons with n equal sides with have order _ for rotational symmetry; 3) What formula do you use to find the smallest angle of rotational symmetry for an octagon?

1) in a figure, the state of being able to be rotated 180 degrees or less about a center point and thereby produce an image that maps precisely onto the original 2) n 3) 360 / n = 360 / 8 = 45

1) pre image: 2) image:

1) the original image before transformation 2) The result of a transformation.

Definition of composition

2 or more transformations performed in sequence to make a single image.

Rules for rotating about origin

R₀,₉₀: (x, y) ↦ (-y, x) R₀,₁₈₀: (x, y) ↦ (-x, -y) R₀, ₂₇₀: (x, y) ↦ (y, -x)

Function notation and mapping rule for translations

Function: T ₐ,b (x, y) Mapping: (x, y) ↦ (x + a, y + b)

What do you do to find the preimage of P after a reflection over the line y=-x?

The rule for a reflection over the line y = -x is (x, y) ↦ (-y, -x). So the rule says to switch them, then negate them. So we do the opposite: negate then switch

What is the coordinate of the vertex of the preimage of Y'' (8,3) when the composition T₋₃, ₇ ◦ R₀,₉₀ made the image?

To undo this, we need to first switch the two transformations. We undo the translation first, then the rotation. To undo the translation, do the opposite. Add 3 to the x-coordinate and subtract 7 from the y-coordinate. To get Y' (11, -4). Then to undo the rotation by going in the opposite direction of the given rule. So instead of using (x, y) ↦ (-y, x), we use (x, y) ↦ (y, -x). To get Y (4, 11).

Definition/conditions and types of non-Rigid Transformation

Transformation that does not preserve shape, size, length, angles, and lines (pre image and image not congruent). Types: dilation and stretch

Definition/conditions and types of rigid transformation

Transformations that preserve shape, size, length, angles, and lines (pre image and image congruent). Types: reflections, translations, rotations

angle of rotation

measure of the angle between a ray drawn from the center of rotation to a point on the pre image and a ray drawn from the center of rotation to the corresponding point on the image

How do you read the composition Rₑ', ₉₀ ◦ rₘ?

rₘ comes first then Rₑ', ₉₀. Do the one to the right first and keep going left.

order of rotational symmetry:

the number of rotations of a figure greater than 0 and no greater than 360 degrees that produce an image identical to the original