12.6 Matrices & Transformations

Graphical transformations (reflections & rotations) can be interpreted using matrices. trans. '

matrix 'x xy y

(point)

general or specific

new point

Def: A 2-by-2 transformation matrix is of the form and it maps each point P(x, y) to its image point P' (x', y')

1 0 1 00 1 0 1x axis y axis

a cb d

T T

Reflection wrt the x-axis Reflection wrt the y-axis

Ex 1) 2 Truths and a LieEach of the following illustrates the transformation of a point.Find and fix the error.

a) b) c)5 7 3; ; ;

3 0 0

1 0 5 5 1 0 7 7 1 0 3 30 1 3 3 0 1 0 0 0 1 0 0

70

x axis y axis x axisT T T

should be

2 2

2 2

1 0 1 00 1 0 1 2 2

( ) ( ) ( ) 2 2

x x xy x x x x

f x x x x x

find

There are also matrices that represent rotations.If you are asked to rotate ° cos sin

sin cosR

We can apply a rotation transformation matrix as well as find out what rotation a combo of transformations is the same as.

We can also use the transformation matrix with an equation.Ex 2) Find and graph the image of f (x) = x2 – x + 2 under Ty-axis.

Ex 3) Find the image. 452, 2 ;

2 2cos 45 sin 45 2 2 22 2sin 45 cos 45 02 2 2 2

2 2

R

Ex 4) Find the transformation formed by the indicated product.1 0 0 1 0 10 1 1 0 1 0

x axis y xT T

cosθ

sinθ

Where does cosθ = 0 and sinθ = 1?at 90°

so, equivalent to R90°

*you will do this in your homework

Homework

#1206 Pg 637 #3–15 odd, 19–23 all, 27, 31, 33, 38, 39