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To summarize what we've been learning so far this year, we have investigated four types of transformations: translations, reflections, rotations, and dilations. The first three types all created congruent figures (since size AND shape stayed the same). Dilations, however, create similar figures because they enlarge or shrink a figure (creating a different size, but still same shape). It is possible for a dilation to lead to a congruent figure, but only if the scale factor is equal to one (since multiplying by one doesn't change side length).
We said that translations can be thought of as a SLIDE. If the figure slides left or right (horizontally) we showed that by either adding or subtracting a value from the x-coordinate (add or subtract based on how many units it slid). It adds to the x-coordinate if it is a positive shift (to the right) and it subtracts from the x-coordinate if it is a negative shift (to the left). If the figure slides up or down (vertically) we showed that by either adding or subtracting a value from the y-coordinate (again, add or subtract based on how many units it slid). We add to the y if it is a positive (up) shift, and we subtract from the y-coordinate if it is a negative (down) shift.
We said that reflections can be thought of as a FLIP. We have only seen a reflection over the x-axis or y-axis so far. If a figure reflects over the y-axis, the y-coordinate should stay the same and the x-coordinate should change signs (positive/negative). We represented this with a rule of (-x , y). If a figure reflects over the x-axis, the x-coordinate should stay the same and the y-coordinate should change signs (positive/negative). We represented this with a rule of (x, -y).
We said rotations can be thought of as a TURN. We need to know how much it turns (we give this in an amount of degrees), we need to know which direction it turned (clockwise or counterclockwise), and we need to know where the center of rotation is (we almost always see it at the origin--(0 , 0)). We did not specifically look at how the coordinates are affected in a rotation, but we do know that rotations do not change shape or size of a figure.
Lastly, we said that dilations can be thought of as either an ENLARGEMENT or a REDUCTION. of a figure. For a dilation, we can almost always assume that the center is at the origin (0 , 0), but we always need to know what the scale factor is. The scale factor is simply the number that is being multiplied to take each original side length to its corresponding dilated side length. For example, if a triangle has been dilated and we know an original side length is 2 while its corresponding side length on the dilated triangle is 8, we know that 4(2)=8, so 4 must be the scale factor. If an original length were 6 and the dilated side length is 2, we know that (1/3)(6)=2, so 1/3 must be the scale factor. Scale factors larger than 1 (like 2, 3.5, 6, 10, etc.) make a figure larger, but scale factors that are between 0 and 1 (like 1/2, 2/3, 3/4, 1/5, etc.) make a figure smaller. As a way of showing the effect of dilations on coordinates, we write it as the scale factor multiplied by both x and y. For example, a dilation with a scale factor of 3 would be written like this: (3x , 3y). That shows that each x and y value must be multiplied by the scale factor in order to get the new coordinate.
There's a lot to remember about transformations, but the things mentioned here are the main ideas. Feel free to look back through any older blog posts for more information and helpful videos about each transformation specifically.