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Transformations: Reflections & Translations
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 We'll learn about REFLECTIONS on a coordinate plane.
 We will also learn about TRANSLATIONS on a coordinate plane.
 And we'll see how this knowledge can help us design a website, create special effects, and even make posters!

Discussion Questions

Before VideoHow many quadrants are in the coordinate plane?ANSWER

Four.

Start at the origin, the intersection of the x and yaxes. Use the xcoordinate to move right or left from the origin, and use the ycoordinate to move up or down.

Find the closest two vertices; the xcoordinate of one point and the ycoordinate of the other form the missing point’s coordinate pair.

The ycoordinates are opposites.

The xcoordinates are opposites.


After VideoWhat are two common types of transformations?ANSWER

Reflections, translations

They both result in an image that is congruent to a preimage.

Reflections make mirror images, while translations slide a shape on the coordinate plane.

Opposites, addition, and subtraction

Use transformations to move (or “map”) one shape onto the other.



Vocabulary

Number line
DEFINE
A horizontal or vertical line used to order numbers from least on the left or bottom to greatest on the right or top.

Origin
DEFINE
The point at (0, 0) on the coordinate plane.

Transformation
DEFINE
Changing a shape, or parts of a shape, from one place to another.

Reflection
DEFINE
Flipping a shape, often over the xaxis or the yaxis.

Translation
DEFINE
Moving a shape from one place to another.

Prime
DEFINE
The notation for a point that is transformed (the symbol ′). Additional marks can be made, such as double prime (′′), triple prime (′′′), and so on, for further transformations.

Preimage
DEFINE
An original shape before it is transformed.

Image
DEFINE
A new shape after it is transformed.

xcoordinate
DEFINE
The first number of a coordinate pair, representing a distance to the left or right of the yaxis.

ycoordinate
DEFINE
The second number of a coordinate pair, representing a distance above or below the xaxis.

Congruent
DEFINE
Shapes that are identical, even if they are in different locations.

Number line
DEFINE

Reading Material
Download as PDF Download PDF View as Separate PageWHAT IS A TRANSFORMATION?A transformation in math is when something moves or changes size. Two common types of transformations are translations and reflections. There are other types of transformations you will learn later.
To better understand transformations: reflections & translations…
WHAT IS A TRANSFORMATION?. A transformation in math is when something moves or changes size. Two common types of transformations are translations and reflections. There are other types of transformations you will learn later. To better understand transformations: reflections & translations…LET’S BREAK IT DOWN!
Two common types of transformations
Two of the most common types of transformations are translations and reflections. In a translation, something moves from one place to another without changing size or orientation. In a reflection, an image flips to another location, just like with a mirror. Transformations can occur in the coordinate plane. Try this one yourself: Think about seeing yourself reflected in a mirror. How does your hair look the same and different from your real hair? Now picture yourself moving (translating) to another part of the mirror. What changed, and what stayed the same?
Two common types of transformations Two of the most common types of transformations are translations and reflections. In a translation, something moves from one place to another without changing size or orientation. In a reflection, an image flips to another location, just like with a mirror. Transformations can occur in the coordinate plane. Try this one yourself: Think about seeing yourself reflected in a mirror. How does your hair look the same and different from your real hair? Now picture yourself moving (translating) to another part of the mirror. What changed, and what stayed the same?Review plotting points on a coordinate plane
The coordinate plane consists of two number lines. The xaxis goes from left to right, and the yaxis goes up and down. The lines meet at the origin, (0, 0). We describe a point on the coordinate plane by its location along these two number lines. The point three units to the right of the origin and five units up is assigned the coordinate pair (3, 5). The point two units to the left of the origin and seven units down is assigned the coordinate pair (–2, –7). Try this one yourself: What coordinate pair represents the point located four units to the right of the origin and six units down?
Review plotting points on a coordinate plane The coordinate plane consists of two number lines. The xaxis goes from left to right, and the yaxis goes up and down. The lines meet at the origin, (0, 0). We describe a point on the coordinate plane by its location along these two number lines. The point three units to the right of the origin and five units up is assigned the coordinate pair (3, 5). The point two units to the left of the origin and seven units down is assigned the coordinate pair (–2, –7). Try this one yourself: What coordinate pair represents the point located four units to the right of the origin and six units down?Reflections can create mirror images
Design software for a website can be set up like a coordinate grid. Before you reflect a graphic, the graphic is called a “preimage,” like the "pretransformed image." After you reflect the graphic, the new graphic is called the “image.” When you reflect a preimage, you have to first choose what line you are going to reflect across, choosing where to put the mirror. To start, you could reflect over the xaxis or the yaxis. When you reflect across the yaxis, the yvalues of all of the coordinates stay the same. The xvalues are opposite the xvalues of the preimage. Just like in a mirror, right becomes left and left becomes right. A triangle with vertices A (4, 7), B (2, 3), and C (5, 1) reflects across the yaxis to A′ (–4, 7), B′ (–2, 3), and C′(–5, 1). The vertices of the image now have ' after their names. A' is read "A prime." Another option is to reflect the preimage over the xaxis. The xvalues of the coordinates stay the same and the yvalues become opposite of the yvalues of the preimage. In this case, the triangle transforms to A′ (4, –7), B′ (2, –3), and C' (5, –1). Try this one yourself: For a preimage with vertices at A (0, 0), B (3, 0), C (3, 3), and D (0, 3), what are the coordinates of the image after the preimage is reflected over the xaxis?
Reflections can create mirror images Design software for a website can be set up like a coordinate grid. Before you reflect a graphic, the graphic is called a “preimage,” like the "pretransformed image." After you reflect the graphic, the new graphic is called the “image.” When you reflect a preimage, you have to first choose what line you are going to reflect across, choosing where to put the mirror. To start, you could reflect over the xaxis or the yaxis. When you reflect across the yaxis, the yvalues of all of the coordinates stay the same. The xvalues are opposite the xvalues of the preimage. Just like in a mirror, right becomes left and left becomes right. A triangle with vertices A (4, 7), B (2, 3), and C (5, 1) reflects across the yaxis to A′ (–4, 7), B′ (–2, 3), and C′(–5, 1). The vertices of the image now have ' after their names. A' is read "A prime." Another option is to reflect the preimage over the xaxis. The xvalues of the coordinates stay the same and the yvalues become opposite of the yvalues of the preimage. In this case, the triangle transforms to A′ (4, –7), B′ (2, –3), and C' (5, –1). Try this one yourself: For a preimage with vertices at A (0, 0), B (3, 0), C (3, 3), and D (0, 3), what are the coordinates of the image after the preimage is reflected over the xaxis?Translations can make an animated superhero fly
To make an animated superhero fly across the screen, a visual effects artist must move the image from one place to another. This is an example of a translation. In a translation, the only thing that changes about the preimage is its location; the angle measurements and side lengths all stay exactly the same. Let’s say a superhero’s head is at (10, 20), and its feet are at (6, 19). If the superhero translates to the right 5 units, only the xcoordinates change. We add 5 to each xcoordinate. The superhero’s head moves to (15, 20), and its feet move to (11, 19). What if, instead of translating to the right, the superhero moved straight down 5 units? In this case, only the ycoordinates change. We subtract 5 from each ycoordinate. The superhero’s head is now at (10, 15) and its feet are at (6, 14). Try this one yourself: A line segment with endpoints at (7, 9) and (5, 11) is translated 4 units to the left. Where are the image’s endpoints?
Translations can make an animated superhero fly To make an animated superhero fly across the screen, a visual effects artist must move the image from one place to another. This is an example of a translation. In a translation, the only thing that changes about the preimage is its location; the angle measurements and side lengths all stay exactly the same. Let’s say a superhero’s head is at (10, 20), and its feet are at (6, 19). If the superhero translates to the right 5 units, only the xcoordinates change. We add 5 to each xcoordinate. The superhero’s head moves to (15, 20), and its feet move to (11, 19). What if, instead of translating to the right, the superhero moved straight down 5 units? In this case, only the ycoordinates change. We subtract 5 from each ycoordinate. The superhero’s head is now at (10, 15) and its feet are at (6, 14). Try this one yourself: A line segment with endpoints at (7, 9) and (5, 11) is translated 4 units to the left. Where are the image’s endpoints?Translations can move in two directions
Let’s say a preimage is a triangle with vertices A (–8, 5), B (–3, 2), and C (–9, 2). A translation three units to the right increases all of the xcoordinates by three, while a translation four units down decreases all of the ycoordinates by four. Vertex A moves from (–8, 5) to become point A′ at (–5, 1). Vertex B moves from (–3, 2) to become point B′ at (0, –2). Vertex C moves from (–9, 2) to become point C′ at (–6, –2).Try this one yourself: If a preimage is a point located at (6, 1), what coordinate pair represents the image after a translation 8 units left and 3 units up?
Translations can move in two directions Let’s say a preimage is a triangle with vertices A (–8, 5), B (–3, 2), and C (–9, 2). A translation three units to the right increases all of the xcoordinates by three, while a translation four units down decreases all of the ycoordinates by four. Vertex A moves from (–8, 5) to become point A′ at (–5, 1). Vertex B moves from (–3, 2) to become point B′ at (0, –2). Vertex C moves from (–9, 2) to become point C′ at (–6, –2).Try this one yourself: If a preimage is a point located at (6, 1), what coordinate pair represents the image after a translation 8 units left and 3 units up?A sequence of transformations can reveal congruence
Two shapes are congruent when they have the same size and shape. If they are congruent, then you can move one on top of the other for an exact match. That is, the shapes overlap perfectly. Imagine a triangle with vertices F (3, 5), G (3, 2), and H (7, 2). Also imagine a second triangle with vertices J (–1, –5), K (–1, –2), and L (3, –2). Can triangle FGH be transformed onto triangle JKL? In this case, start by reflecting triangle FGH over the xaxis by changing the ycoordinates to their opposites. That gives us points F′ (3, –5), G′ (3, –2), and H′ (7, –2). Then translate this triangle 4 units to the left by subtracting 4 from all xcoordinates. Point F′ moves over to where J is at (–1, –5), point G′ moves over to where K is at (–1, –2), and point H′ moves over to L at (3, –2). There you have it! Since the vertices overlap perfectly, and none of the angles or side lengths changed at all, triangle FGH and triangle JKL are congruent: an exact match! Try this one yourself: Draw two identical triangles in the coordinate plane, where getting to the image requires at least one reflection and one translation. Identify the vertex coordinates of each triangle.
A sequence of transformations can reveal congruence Two shapes are congruent when they have the same size and shape. If they are congruent, then you can move one on top of the other for an exact match. That is, the shapes overlap perfectly. Imagine a triangle with vertices F (3, 5), G (3, 2), and H (7, 2). Also imagine a second triangle with vertices J (–1, –5), K (–1, –2), and L (3, –2). Can triangle FGH be transformed onto triangle JKL? In this case, start by reflecting triangle FGH over the xaxis by changing the ycoordinates to their opposites. That gives us points F′ (3, –5), G′ (3, –2), and H′ (7, –2). Then translate this triangle 4 units to the left by subtracting 4 from all xcoordinates. Point F′ moves over to where J is at (–1, –5), point G′ moves over to where K is at (–1, –2), and point H′ moves over to L at (3, –2). There you have it! Since the vertices overlap perfectly, and none of the angles or side lengths changed at all, triangle FGH and triangle JKL are congruent: an exact match! Try this one yourself: Draw two identical triangles in the coordinate plane, where getting to the image requires at least one reflection and one translation. Identify the vertex coordinates of each triangle. 
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Draw a square with vertices at (3, 5), (3, 9), (7, 5), and (7, 9). Reflect the square over the xaxis. What are the coordinates of the new vertices?
Draw a triangle with vertices at (–5, –6), (–2, –2), and (–1, –3). Translate the triangle 4 units right and 1 unit down. What are the coordinates of the new vertices?
Triangle ABC has vertices at (1, 2), (4, 2), and (4, 6). Triangle JKL has vertices at (–1, –1), (–4, –1), and (–4, 3). What sequence of transformations maps triangle ABC onto triangle JKL?
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