# Transformation Investigations

- Author:
- Jennifer Bayley, Heidi Moore

**Coordinate Transformation Investigation**Use the following applet to investigate the coordinate changes under reflection, rotation, translation and dilation transformations on a coordinate grid. You will be able to manipulate the original triangle ABC to assist in your investigating.

*Note:*

- Since all images (newly created triangles) use A', B' and C', they are subscripted with numbers.
- All rotations are about the origin and are counterclockwise.
- All dilations use the origin as the center of dilation.

**Instructions: Answer all of the questions. You may either type your answers in a text editor and then email it to me, or you may write your responses in notebook paper.**

**1.**What happens to the coordinates of the vertices of the triangle after a reflection across the y-axis? Do the following steps and decide if your hypothesis still holds true:

**a.**Move one vertex of the original triangle onto the y-axis.

**b.**Move two vertices of the original triangle onto the y-axis.

**c.**Move the original triangle so that it crosses the y-axis.

**Reflection across the y-axis Rule: (x, y) ==> (________, ________)**

**2.**What happens to the coordinates of the vertices of the triangle after a reflection across the x-axis? Do the following steps and decide if your hypothesis still holds true:

**a.**Move one vertex of the original triangle onto the x-axis.

**b.**Move two vertices of the original triangle onto the x-axis.

**c.**Move the original triangle so that it crosses the x-axis.

**Reflection across the x-axis Rule: (x, y) ==> (________, ________)**

**3.**What happens to the coordinates of the vertices of the triangle after a reflection across the line y=x? Do the following steps and decide if your hypothesis still holds true:

**a.**Move one vertex of the original triangle onto the line y=x.

**b.**Move two vertices of the original triangle onto the line y=x.

**c.**Move the original triangle so that it crosses the line y=x.

**Reflection across the line y=x Rule: (x, y) ==> (________, ________)**

**4.**Under a reflection, do the sides of a triangle maintain their lengths?

**5.**What happens to the coordinates of the vertices of the triangle after a rotation of 90º? Do the following steps and decide if your hypothesis still holds true:

**a.**Move one vertex of the original triangle so that it is on the origin.

**b.**Move the original triangle so that the origin is within the triangle.

**Rotation of 90º Rule: (x, y) ==> (________, ________)**

**6.**What happens to the coordinates of the vertices of the triangle after a rotation of 180º? Do the following steps and decide if your hypothesis still holds true:

**a.**Move one vertex of the original triangle so that it is on the origin.

**b.**Move the original triangle so that the origin is within the triangle.

**Rotation of 180º Rule: (x, y) ==> (________, ________)**

**7.**What happens to the coordinates of the vertices of the triangle after a rotation of 270º? Do the following steps and decide if your hypothesis still holds true:

**a.**Move one vertex of the original triangle so that it is on the origin.

**b.**Move the original triangle so that the origin is within the triangle.

**Rotation of 270º Rule: (x, y) ==> (________, ________)**

**8.**Under a rotation, do the sides of a triangle maintain their lengths?

**9.**Before beginning the translation part of your investigation, note that the purple arrow to the left is known as a translation vector. The ordered pair next to it represents the move in the vertical, or x, direction and the horizontal, or y, direction. What happens to the coordinates of the vertices of the triangle after a translation? Do the following steps and decide if your hypothesis still holds true:

**a.**Move the translation vector so that it is horizontal.

**b.**Move the translation vector so that it is vertical.

**c.**Move the translation vector so that it includes negative numbers.

**Translation Rule: (x, y) ==> (________, ________)**{use a horizontal move,

*h*, and a vertical move,

*v*, which is also written as <

*h*,

*v*>}

**10.**Under a translation, do the sides of a triangle maintain their lengths?

**11.**For the dilation part of your investigation, you need to know that the

*s*represents the scale factor. To begin, set

*s*= 2. What happens to the coordinates of the vertices of the triangle after a dilation? Do the following steps and decide if your hypothesis still holds true:

**a.**Move the triangle so that at least one vertex is on an axis.

**b.**Move the original triangle so that the origin is within the triangle.

**c.**Change the scale factor to 3 (you may need to zoom out to see the entire image).

**d.**Change the scale factor to 0.5.

**Dilation Rule: (x, y) ==> (________, ________)**{use a scale factor of

*s*}

**12.**Under a dilation, do the sides of a triangle maintain their lengths?

**13.**Within the dilation option, now change the scale factor to 1 and observe what happens. Why does this happen?

**14.**If the scale factor of a dilation is 2, do you think that would be considered a reduction or an enlargement?

**15.**If the scale factor of a dilation is 0.5, do you think that would be considered a reduction or an enlargement?

**16.**By looking only at the scale factor of a dilation, how can you tell if it is a reduction or an enlargement?