When will I use Transformations?

I have had a lot of you asking:
"What good are transformations to me?"
"When will I use this information!"

Computer animations, especially something like the animated cartoon movie Toy Story or Monster's Inc., all use transformations of their shapes to get the figures to move around on the screen. Of course, these artists who create the motion pictures are allowed to use computers that help them with these mathematical calculations. We are learning the basic background of being able to move an object in space. Putting calculations together, we could create a computer animation using these transformation rules.

Please check out this site on 3D computer animation.
http://computer.howstuffworks.com/3dgraphics7.htm

Magnitude/Size Change Transformation

Given a magnitude, m, the size transformation will follow the following rule:
(note: m can be any number)

(x,y) -> (mx,my)

multiply the magnitude by both your x and y coordinates to get the new transformed image.

A magnitude/size transformation is a Non-Rigid transformation because the size changes shapes in the following ways:

1. The lengths between the pre-image and transformed image are multiplied by the magnitude.

2. The slopes between the sides of the pre-image are the same as the slopes between the corresponding transformed image sides.

3. The shapes are the same shapes just different size.

4. The area of the pre-image is multiplied by a factor of the magnitude squared to get the transformed image area size.

Please refer to the Master Worksheet 44 that was given in class for more information on this topic. Also, page 128 in your text, #4 a,c,d.

Rotation Transformations

Rules for rotating images:

90 degree clockwise rotation = 270 degree counter-clockwise rotation
(x,y) -> (y,-x)

180 degree clockwise rotation = 180 degree counter-clockwise rotation
(x,y) -> (-x,-y)

270 degree clockwise rotation = 90 degree counter-clockwise rotation
(x,y) -> (-y,x)

Translations and Reflections

Rules for Creating the Transformations: Translations and Reflections

Translation with components h and k.
(x,y) -> (x+h, y+k)

Reflection across the x-axis
(x,y) -> (x, -y)

Reflection across the y-axis
(x,y) -> (-x, y)

Reflection across the line y = x
(x,y) -> (y, x)

Reflection across the line y = -x
(x,y) -> (-y, -x)

Calculator Programs

You can program your calculator to calculate slope, distance, and midpoint for you. You can also get these programs by linking your calculator to another calculator that already has the programs.

SLOPE

:ClrHome
:Input "X COORD?",A
:Input "Y COORD?",B
:Input "X COORD?",C
:Input "Y COORD?",D
:"(B-D)/(A-C)->L
:Disp "SLOPE IS",L
:Disp "PERP SLOPE IS",-1/L
:Stop

Note: -> is the STO button on your calculator.

DISTANCE

:ClrHome
:Input "FIRST X VALUE?",A
:Input "FIRST Y VALUE?",B
:Input "SECOND X VALUE?",C
:Input "SECOND Y VALUE?",D
:square root((A-C)^2+(B-D)^2)->L
:Disp "DISTANCE IS",L
:Stop

Note: square root needs to be your square root button, not the words square root

MIDPOINT

:ClrHome
:Input "X COORD?",A
:Input "Y COORD?",B
:Input "X COORD?",C
:Input "Y COORD?",D
:(A+C)/2->X
:(B+D)/2->Y
:Disp "MIDPT COORDS"
:Disp "X VALUE",X
:Disp "Y VALUE",Y
:Stop

Note: -> means the STO button on your calc.

Polygon Properties

Proving shapes to be polygons by looking at their properties and not their visual appearance is important. Make sure you know all the polygon properties and how to determine if they are what they look like!

PERPENDICULAR SIDES: For sides to be perpendicular their slopes must be opposite reciprocals
(for example: if there is a slope of -4/5 the opposite reciprocal is 5/4)
(for example: if there is a slope of 8 the opposite reciprocal is -1/8)

PARALLEL SIDES: For sides to be parallel their slopes must be the same

CONGRUENT SIDES: Congruent means they are the same length

SQUARE
Quadrilateral
Adjacent sides are perpendicular
All sides are equal length
Opposite sides are parallel

RECTANGLE
Quadrilateral
Adjacent sides are perpendicular
Opposite sides are equal length
Opposite sides are parallel

RHOMBUS
Quadrilateral
All sides are equal length
Opposite sides are parallel

PARALLELOGRAM
Quadrilateral
Opposite sides are parallel
Opposite sides are equal length

KITE
Quadrilateral
2 pairs of adjacent sides are of equal length
Diagonals are perpendicular

TRAPEZOID
Quadrilateral
1 set of opposite sides are parallel

RIGHT TRIANGLE
3 sides
2 of the sides are perpendicular

ISOSCELES TRIANGLE
3 sides
2 of the sides are of equal length

EQUILATERAL TRIANGLE
3 sides
All sides are equal length

Finding Midpoint of a Segment between 2 Points

Given 2 points

(a,b) and (c,d)

use this formula to find the midpoint of the segment between the two points.

x-value of the midpoint: (a+c)/2

y-value of the midpoint: (b+d)/2

Note: Each of these needs to be calculated separately and then put into a coordinate point.

Midpoint: ((a+c)/2,(b+d)/2)


Example: Find the midpoint between the points A(-12,20) and B(5,-18)

x-value of the midpoint = (-12+5)/2
x-value of the midpoint = (-7)/2
x-value of the midpoint = -3.5

y-value of the midpoint = (20+-18)/2
y-value of the midpoint = (2)/2
y-value of the midpoint = 1

Midpoint: (-3.5,1)

Slope of Segment Between Points

Slope = Rise/Run

Given two points

(a,b) and (c,d)

use this formula to find the slope of the segment between the two points.

Slope = (d-b) /(c-a)

Example: Find the slope between the points A(-12,20) and B(5,-18)

Slope = (-18-20) /(5--12)

Slope = (-38) /(17)

Slope approximately = -2.235 or -2.24

Distance Between Points

Given 2 coordinate points:

(a,b) and (c,d)

To find the distance between the points follow this formula.
Note: Sometimes your calculator is not as smart as you are and so I advise you to work through the problem as far as you can without the use of your calculator.

Take the Square Root of the quantity ((a - c)^2 + (b - d)^2)

Note: ^2 means square the quantity.

Example:

A(-12,20)
B(5,-18)

Square Root of the quantity ((-12 - 5 )^2 + ( 20 - -18 )^2)

Square Root of the quantity ((-17)^2 + (38)^2)

Square Root of the quantity (289 + 1444)

Square Root of the quantity (1733)

approximately = 41.629 or 41.63

Note: Please round your answers to at least the 2nd decimal place.

Solving Systems of Equations Test

To prepare for the test, please review all 4 methods of how to solve systems of equations and how to set up 2 linear equations when given a scenario.

Please come in before or after school on Monday for extra help. The test is on Tuesday October 4th.

Sample Problem:

Try this problem and check your answers with the solution at the bottom.


The solutions should all be the same for each square. This is because we are solving the same problem in 4 different ways. So the solution for x and the solution for y will be the same throughout.