Geo Chapter 14: Transformations

mapping

aka a function. every input has one output

how to find the image of x strategy

plug in the x value

dilation

a transformation related to similarity, not congruence. smaller → bigger or vice versa

reflection definition

a) is an isometry b) preserves distance, angle measure and area-ONLY ORIENTATION IS CHANGED c) like a direct mirror image

finding the preimage of (x,y)

a) usually an equation will be given such as: {L : (x,y) → (2x - 1, 2y + 3)} b) if (example) point (x,y) was (9,2), then solve [2x - 1 = 9] and [2y + 3 = 2] to find the preimage values - to find the image of (x,y), then plus in the values of (x,y) ex: 2(9) - 1, 2(2) + 3

one-to-one mapping

every input must have one output. in the contexts of images, every image must has a preimage and vice versa

glide reflections

every point P is translated or glided over along the x axis, then reflected over the x axis from its translated/glided position

formula for reflection

for a reflection in line m = Rm if Rm maps A to A': [Rm: A → A'] or [Rm(A) = A']

how to find the image of x

function: g(x) = 2x + 6 x= 7 g(7) = 2(7) + 6 = 20 the image of 7 is 20.

how to find the preimage of x

function: g(x) = 2x + 6 x= 8 g(x) → 2x + 6 = 8 x= 1, so the preimage of 8 is 1.

isometry definition

if a transformation maps every segment to a congruent segment aka EXACTLY equal segments, angles and shapes are made

translation definition

maps any point (x, y) to (x+a, y+b) where a and b are constants. in a graph, moves it up or down, in a shape, moves it up or down, lengthens segments formula: [T: (x, y) → (x+a, y+b)]

transformation definition

one-to-one mapping from the whole plane to the whole plane

preimage and image

preimage: the original image: the copy context: P is the preimage of 'P, and 'P is the image of P

how to find the preimage of x strategy

solve FOR the x value in the function. function = preimage

how should you test if something is in isometry?

use the distance formula

rotation

when a rotation takes place of a certain # of degrees, from the original point of O, a new point of 'P is made

what is an example of one-to-one mapping?

yes: f : x → 2x, because there is only one preimage per image no: f : x → x², because squares, such has 9, can have two preimages of 3 and -3