Question

Using a pre-image, you should be able to perform a translation and a rotation and write mathematical notation for each. Show the transformation and write the function rule for a translation if you want to move the preimage seven units to the left and two units up?

a) Translation Transformation:
T(x,y)→(x−7,y+2)

b) What do you need to know in order to do a rotation? Complete a rotation of your choice and identify the rule for this.

a) The center of rotation and the angle of rotation
b) The coordinates of the preimage
c) The translation rule
d) The scale factor for the rotation

Answer 1

A translation that moves an object seven units left and two units up is described by the function T(x, y) → (x - 7, y + 2). For rotation about the origin by an angle φ in radians, the coordinates of the rotated point are given by x' = x cos(φ) - y sin(φ), y' = x sin(φ) + y cos(φ). To verify the rotation, ensure that the distance from the origin is invariant before and after the transformation.

Step-by-step explanation:

To perform a translation transformation, you need to move the pre-image a certain number of units horizontally (left or right) and vertically (up or down). For a translation that moves seven units to the left and two units up, the function rule is T(x, y) → (x - 7, y + 2).

For a rotation, you need to know the center of rotation and the angle of rotation. The rotation rule depends on these parameters. If we choose the origin (0,0) as the center and rotate by an angle φ, the rule for a counterclockwise rotation is x' = x cos(φ) - y sin(φ), y' = x sin(φ) + y cos(φ). The coordinates of the pre-image should be known, and the angle of rotation should be in radians.

To check if the answer is reasonable, you can verify that the distance of a point from the origin remains the same after a rotation by showing that the squared distance (using the Pythagorean theorem) is invariant: x'² + y'² = x² + y².