Transformations on the Coordinate Plane Mr. J. Grossman

Transformation… a definition  A transformation is a general term for four specific ways to manipulate the shape of a point, a line, or two-dimensional figure on the coordinate plane.  The original shape of the object is called the pre-image and the final shape and position of the object is the image under the transformation. pre-imageimage

Transformation Types  Translation Translation  Reflection Reflection  Rotation Rotation  Dilation Dilation  A composition of transformations means that two or more transformations will be performed on one object. For instance, we could perform a reflection and then a translation on the same point.composition of transformations reflectiontranslation

Transformation & Congruence  An isometry is a transformation that preserves mathematical congruence. In other words, it is a transformation in which the preimage and image have the same side lengths and angle measurements. congruence  The following transformations maintain their mathematical congruence. Translations (a translation is considered a 'direct isometry' because it not only maintains congruence, but it also, unlike reflections and rotations, preserves its orientation. Translations Rotations Reflections

Transformation & Congruence  On the other hand,a dilation is not an isometry because its image (most times) is not congruent with its pre-image.dilation  The preimage and image do not necessarily have the same side lengths and angle measurements.

Translations  Definition: A translation is a transformation of the coordinate plane that moves every point in the plane the same distance in the same direction. ... without rotating, resizing or anything else, just moving!

Translations  A translation "slides" an object a fixed distance in a given direction. The original object and its translation have the same shape and size, and they face in the same direction. A translation creates a figure that is congruent with the original figure and preserves distance (length) and orientation (lettering order). A translation is a direct isometry.

Translations  Properties preserved… Distance (lengths of segments are the same) Angle measures (remain the same) Parallelism (parallel lines remain parallel) Colinearity (points stay on the same lines) Midpoint (midpoints remain the same in each figure) Orientation (lettering order remains the same)

Translation Notation Translation of h, k: The -7 tells you to subtract 7 from all of your x-coordinates, while the -3 tells you to subtract 3 from all of your y-coordinates. This may also be seen as T-7,-3(x,y) = (x -7,y - 3).

Translation Notation  Translation Notation is also known as mapping:  This is read: "the x and y coordinates will be translated into x - 7 and y - 3". Notice that adding a negative value (subtraction), moves the image left and/or down, while adding a positive value moves the image right and/or up.)

Translations Examples  Notice how each vertex moves the same distance in the same direction.

Translations Examples  In this example, the "slide" (translation) moves the figure 7 units to the left and 3 units down.

Practice…  Under the translation, the point (2,5) becomes…  Points E (3, 1), F (–2, 0), and G (1, –3) are the vertices of Δ EFG. Draw the image and list the coordinates of Δ E'F'G' after the translation (x, y)  (x + 5, y + 7).

Practice…  Given the translation: (x, y )  (x + 4, y + 6) If point Q has the coordinates (–4, 0), find the coordinates of point Q'.  Given the translation: (x, y )  (x + 4, y + 6) If point R' has the coordinates (1, –2), find the coordinates of point R.

Any questions?  If not…