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Transformations: Rotations & Dilations
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What you will learn from this videoWhat you will learn
- We’ll learn that dilations make shapes larger and smaller.
- We will also learn that rotations turn shapes around a point on a coordinate plane.
- And we'll see how this knowledge can help us make art, edit photos, and even solve a mystery!
- Discussion Questions
Before Video
What is a translation?ANSWERA translation is a horizontal or vertical shift of a point or figure. The image produced by translation is congruent to the original.
A reflection is a mirror image of a point or figure taken across a reflection line. The reflections are usually across the x- or y-axis of a coordinate plane.
Yes; reflecting a figure does not change its angles or side lengths, so the image is congruent to the preimage.
Points are defined using an ordered pair (x, y) so that the first number is the distance from the origin on the x-axis, and the second is the distance from the origin on the y-axis.
Yes, I can perform multiple transformations to get an image. The order that I perform the transformations in matters, so I need to follow the order of transformations. This is similar to the order of operations. The order is rotations, reflections, dilations, and then translations.
After Video
What is a scale factor?ANSWERA scale factor is the value I multiply the coordinates of the original point by to get its image point in a dilation.
Enlargements and reductions are both dilations, but an enlargement produces an image that is larger (farther from the origin) than the original, and a reduction produces an image that is smaller (closer to the origin). Enlargements have scale factors larger than 1, and reductions have scale factors between 0 and 1.
First, I need to know whether to rotate clockwise or counterclockwise. The mapping rule for a clockwise 90° rotation is (x,y)→(y,-x). If I want to rotate counterclockwise, I need to find the equivalent angle: 270° clockwise is the same as 90° counterclockwise. The mapping rule for a clockwise 270° rotation is (x,y)→(-y,x).
First, find the corresponding vertex points. Then determine if the image is in the same orientation or a different one. If the orientation is different, look for a rotation or reflection. If the orientation is the same, check if both figures are the same size. If they are not the same size, then a dilation was used. Divide the image coordinates by the corresponding original coordinates. The answer gives the scale factor for the dilation. If there was no dilation, then the difference between the image coordinates and the original coordinates gives the translation.
Two 2D figures are similar if all the corresponding angles are equal, and all side lengths share the same scale factor between them. Translations, reflections, dilations, and rotations all create similar shapes.
- Vocabulary
- Coordinate plane DEFINE
A grid made of two number lines that cross one another at zero.
- x-axis DEFINE
The horizontal number line, going left to right.
- y-axis DEFINE
The vertical number line, going down to up.
- Origin DEFINE
The point where the x- and y-axes cross, (0, 0).
- Point DEFINE
A point on a coordinate plane is represented by two numbers separated by a comma, in parentheses. In the point (x, y), "x" represents the horizontal distance from the origin, and "y" represents the vertical distance from the origin.
- Transformation DEFINE
The process of changing a figure or expression into its "image" by following a mathematical rule. Performing transformations is called mapping, and the rule used is called a mapping rule.
- Image DEFINE
The figure or expression produced by performing a transformation on the original figure or expression.
- Dilation DEFINE
A transformation that makes something larger or smaller. In a dilation, multiply all coordinates of the preimage by a scale factor to produce the image.
- Translation DEFINE
A horizontal or vertical shift of the preimage to produce the image. Horizontal shifts are achieved by adding a value to the x-coordinate of all points in the original. Vertical shifts are achieved by adding a value to the y-coordinate of all points in the original.
- Reflection DEFINE
A mirror image of the original across the line of reflection. To reflect across the x-axis, multiply all y-coordinates by –1. To reflect across the y-axis, multiply all x-coordinates by –1. To reflect over the line y = x, switch the x- and y-coordinates. To reflect across the line y = –x, switch the x- and y-coordinates and multiply both by –1.
- Scale Factor DEFINE
The factor used to dilate an original shape to its image. The scale factor is how many times larger the image is than the original. Scale factors are positive values, with scale factors less than 1 producing an image smaller than the preimage, and scale factors larger than 1 producing images larger than the preimage.
- Rotation DEFINE
Turning a point or figure about a point to find the image. Rotations are commonly about the origin, where a point (x, y) rotated 90° clockwise has an image of (y, –x), 180° has an image of (–x, –y), and 270° has an image of (–y, x).
- Coordinate plane DEFINE
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