Unit 1 Notes Flashcards
Mapping
same thing as Function. A correspondence of A to B IF AND ONLY IF each member of A corresponds to one and only one member of B
Transformation
When you have the same number of points in the pre-image and image. One to one function.
Isometric transformation (Rigid motion)
A transformation that preserves the distances and/or angles between the pre-image and image
Non-isometric transformation (Non-rigid motion)
A transformation that does not preserve the distance and angles between the pre-image and image
Stretch
Non-isometric transformation
Where one dimension’s scale factor is different than the dimension’s scale factor. A strecth definitely distorts the shape
Dialation
Where both dimension’s scale factors are the same. The shape is proportional, not identical. Dialation changes the size of the shape making it a NON-ISOMETRIC transformation.
what does to carry a shape onto itself mean
That the shape has symmetry
Three types of symmetries
Line symmetry, rotation symmetry, and point symmetry
Line of symmetry definition
A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in the line
How many lines of symmetry do rectangles have
2
Why is the maximum lines of symmetry that a polygon can have equal to it’s number of sides
Because all sides and all angles are congruent
Rotational symmetry
A geometric figure has rotational symmetry if the figure is the image of itself under a rotation about a point through any angles who measure is strictly between 0 and 360. 0 and 360 are excluded from counting as having rotational symmetry because it represents the starting position. You can tell if a figure has at least an order of 2 if you flip the image upside down and it looks the same
Angle of rotation
The smallest angle through which the figure can be rotated to coincide with itself. This number will always be a factor of 360. To find divide 360 by the order of rotation
Order of rotation symmetry
The number of positions in which the object looks exactly the same. When determining order, the last rotation returns the object to its original position. Order 1 implies no true rotational symmetry since a full 360 degree rotation was needed.
Point symmetry
Exists when a figure is built around a point such that every point in the figure has a matching point that is:
the same distance from the central point
But in the opposite direction.
- Turn the figure upside down and see if it looks the same to see if it has point symmetru.
- All have an order of 2 and even orders
Reflection
A reflection over a line is an isometric transformation in which each point on the original figure has an image that is the same distance from the line of reflection has the original point but in on the opposite side of the line.
- The image does not change size or shape
- AA’ ll BB’ ll CC’
AA’ is not congruent to BB’ or CC’
Isometric properties of a reflection
The following stay the same
-Distance (lengths of the segments are the same)
- Angle measure (angles stay the same)
-Parallelism (things that were parallel are still parallel)
- Collinearity (points on a line, remain on the line)
After a reflection, the pre-image and image are identical
Transformation properties of a reflection
The following change:
- Distance: point son the plane move different distances, depending on their distance from the line of reflection. Points further away from the line of reflection move a greater distance than those closer to the line of reflection
- Orientation: is reversed (orientation is the order of the points about the shape)
- Special points: Points on the line of reflection do not move at all under the reflection (A + A’).
Rotation
An isometric transformation that turns a figure about a fixed point called the center of rotation. (A can = A’ if the point is on the center of rotation). AA’ is not ll to BB’
and AA’ is not congruent to BB”
Angle of rotation
Rays drawn from the center of rotation to a point and it’s image
Rotation direction
Counter clockwise- positive
Clockwise- negative
Co-terminal angles
Because angle are formed along an arc of a circle there are two ways to get to the same location, a positive direction and a negative direction.
The co-terminal angle formula finds the other positive or negative angle
Coterminal angle= initial angle+ 360 n and n is an integer like 1 or -1
Isometric properties of a rotation
The following stay the same:
-Distance (lengths of segments are the same)
-Angle measure (angles stay the same)
-Parallelism (things that were parallel are still parallel)
-Collinearity (points on a line, remain on the line)
After a rotation, the pre-image and image are identical
Transformation properties of a rotation
The following change
- distance: points in the plane move different distances, depending on their distance from the center of rotation. A point father away from the center of rotation maps a greater distance than those points closer to the center of rotation
- Orientation is the same
- Special points: the center of rotation is the only point in the plane that is unchanged
Translation
Slides an object a fixed distance in a given direction. A vector is used to describe the fixed distance and the given direction.
AA” ll BB” (a fixed idrection)
AA” is congruent to BB” (a fixed distance)
Isometric properties of a translation
- Distance
- Angle measure
- Parallelism
- Collinearity (point on a line, remain on a line)
Transformation properties of a translation
- Distances are the same: points in the plane all map the exact same distance AA’ is equal to BB”
- Orientation is the same
- Special points: no special points, all points move
Translated segments
All segments that are translated are parallel to each other
Reflection of Y axis rule
Ryaxis (x,y) = (-x,y)
Reflection over the x axis rule
Rxaxis (x,y) = (x,-y)
Reflection over Y=x rule
Ry=x (x,y) = (y,x)
Rotation of 90 degree rule
RO,90(x,y) = (-y,x)
Rotation of 180 degree rule
(-x,-y)
Rotation of 270 degrees rule
(y,-x)