2Transformations in the Coordinate Plane Unit 5Transformations in the Coordinate PlaneEssential Question:How do we translate a geometric figure in the coordinate plane?
3A transformation is a change in a geometric figure’s position, shape, or size. 4 Main Types of TransformationsTranslationsReflectionsRotationsDilations
4Translation- a transformation that moves each point of a figure the same distance in the same direction.The shape, size, or orientation does NOT change. So we say the figures are congruent.Polygon- shape made of straight lines and is closed on all sides.
5Practice: Translate each figure as instructed below Practice: Translate each figure as instructed below. Describe using words (up, down, left, right) how the figures were translated. 1. (𝒙,𝒚) (𝒙+𝟔,𝒚) 2. (𝒙,𝒚) (𝐱 ,𝐲 −𝟔)
7Describe the transformation to the green hexagon that will produce the BLUE hexagon.
8A dilation is a transformation that changes the size or the shape of a figure but making it larger or smaller. The notation for a dilation is similar to a translation, however, a dilation will always multiply or divide the variables x and y by something. Example (x,y) (3x+2, y) (x,y) (2x-3, 3y+1)
9Homework: Graph the following figure using the given coordinates Homework: Graph the following figure using the given coordinates. Perform the transformation. Tell whether the transformation is a translation or a dilation. 1) (x,y) (x-5, y-3) 2) (x,y) (3x+1, 2y) A (1,1) A (-3,1) B (2,1) B (-3,-1) C (1,3) C (-2,1) D (-2,-1) 3) (x,y) (x+1, y+4) 4) (x,y) (2x+1, 3y-3) A (2,-1) A (-1,2) B (1,-2) B (-1,1) C (2,-3) C (0,1) D (3,-2) D (0,2)
10The next type of transformation we will look at is called a reflection The next type of transformation we will look at is called a reflection. A reflection flips an object or shape across a specific line.Example:Practice: Reflect the image below as instructed.1. Reflect over the x-axis 2. Reflect over the y-axis
11Examples: of reflecting over a horizontal or vertical line While reflecting over the x or y axis is a very common way to reflect objects. We can also reflect shapes over any line (not just an axis).Examples: of reflecting over a horizontal or vertical line1. Reflect over 𝒙=𝟑 2. 𝑹𝒆𝒇𝒍𝒆𝒄𝒕 𝒐𝒗𝒆𝒓 𝒚=−𝟏Example of reflecting over a non-horizontal or non-vertical line.3. Reflect over 𝒚=𝒙Notice that the coordinates (x,y) from the first shape end up being flipped (y,x) in the second shape.When reflecting over the line 𝑦=−𝑥 you flip the x and y and change the sign.
12practice 1. reflect over 𝒚=−𝟐. 2. reflect over 𝒙=𝟑 3. Reflect over 𝒚=𝒙 practice 1. reflect over 𝒚=−𝟐 2. reflect over 𝒙=𝟑 Reflect over 𝒚=𝒙 4. reflect over 𝒚=−𝒙
13The last type of transformation we will look at is called a rotation The last type of transformation we will look at is called a rotation. A Rotation is a transformation that turns a figure about a fixed point.There are 3 important things we need to know before rotating an objectPoint of rotationAngle of rotationDirection of rotationExample:The point of rotation is usually the origin, but sometimes might be a point on the shape.The angle of rotation will be in increments of 90˚ (90˚, 180˚, 270˚)The direction will either be clockwise (the way a clock turns) or counter clockwise (the opposite way a clock turns).
14Example: Rotate 90˚ clockwise about the origin. 90̊ = 1 turn Tip: When rotating you can turn the paper sideways depending on which direction and how many degrees the rotation is. Then copy down the coordinates looking at the graph the way it is, and turn it back to its original position and graph the new coordinates.Example: Rotate 90˚ clockwiseabout the origin.90̊ = 1 turn180̊ = 2 turns270˚ = 3 turnsclockwise = rightcounter-clockwise = left90˚ turn clockwise180˚ turn clockwise270˚ turn clockwise
15Practicerotate 90̊ clockwise about the origin 2. rotate 90˚ counter-clockwiseabout the origin3. rotate 180˚ about the origin 4. rotate 270˚ clockwise about the origin
16An isometry a transformation to an object or shape that preserves distances between the points of the shape. A rotation, reflection or translation are all examples of an isometry, since the distances between two points on the plane remain the same afterwards.A dilation increases or decreases the size of a shape or object. A dilation is NOT an isometry because distance between points is not preserved.Example of a dilation(𝒙,𝒚) (𝒙+𝟓, 𝟐𝒚)Because of the dilation to the y values (2y) you can see the distances between the points does not stay the same. Therefore this is not an isometry.
17How to rotate around a point that is NOT the origin Example: Rotate 90˚ counter clockwise around the point (−1,2)1st: Graph the point thatwe want to rotate around.2nd: Going through the point draw a horizontal line, and a vertical line.3rd: Now rotate the paper and use the new lines as the x and y axis to reference where our new shape goes.4th: Graph the new shape
18Practice1) 180˚ around the point (−𝟐,−𝟐) 2) 90˚ CCW around the point (−𝟑,−𝟐)3) 270˚ CW around the point (𝟎,𝟏) 4) 90˚ CW around the point (𝟐,−𝟏)
19Warm-up: Perform all the transformations in order Warm-up: Perform all the transformations in order. Each time using the new shape as the object you are transforming. reflect over the line 𝐲=𝟏 then (𝐱,𝐲) (𝐱−𝟕,𝐲+𝟏) Rotate around the origin 90˚ cw
20This is clearly a translation: (x,y) to (x+4, y+4) Mapping is the description of the transformation that puts one given shape exactly on top of another given shape. Example: What transformation maps the shaded figure onto the non-shaded figure.Tips: RememberA translation will not change the orientation of the shape. It will simply move the shape up, down, left and right.A reflection flips a shape across a line changing its orientation from being flat to tall or from facing one direction to facing the opposite.A rotation or 90˚ or 270˚ will cause the shape to turn onto its side. A rotation of 180˚ will flip a shape upside down (however, with squares and rectangles you cant tell).Answer:This is clearly a translation:(x,y) to (x+4, y+4)
21Practice: Describe the transformations that maps the shaded figure onto the non shaded figure. 3.