04 Introduction to Analytic Geometry
College Algebra
4.1 Coordinate Systems
OriginLinePlanePointCoordinate (a,b) same as (x,y)UnitsThree Space (a,b,c)
Coordinate System Reviewy
x
III
III IV
Origin
Ordered Pairs Review : (a,b)
b
a
III
III IV
(a,b)(-a,b)
(a,-b)(-a,-b)
Transformations & Coordinate Geometry
Transformations – Model Motion
Translation – Glide or SlideRotation – (about an axis)Reflection – Mirror imageDilation – larger or smaller
Terminology
Image – final image after transformationLabeled with “Prime”Pre-image – image before transformationLabeled with Capital Letters
A A’
B B’
Pre-Image Image
Horizontal Translation
Translation
Pre-ImageImage Slide Arrow
A
B CA’
B’ C’
Rotation – 90 180 270 45
Pre-Image
Image
90
Image180
Imag
e27
0
Note: Example Rotation is Clockwise
Graphing Motion
( x , y )( x , y )( x , y )( x , y )( x , y )( x , y )
( x + h, y ) ( x , y + v ) ( x , -y ) ( -x , y ) ( -x , -y ) ( nx, ny )
Pre-Image ImageHorizontal Translation
Vertical Translation
Reflection through x-axis
Reflection through y-axis
180 Rotation about Origin
Dilation
Back to the text…
Distance Formula1-D2-D3-D
2-D: “THE” Distance formula
what do you know about the distance formula???
2-D: “THE” Distance formula
A
B
2-D: “THE” Distance formula
A
B
2-D: “THE” Distance formula
(-3,2)
(5,7)
d = sqrt((5- -3)2 + (7-2)2)
2-D: “THE” Distance formula
(-3,2)
(5,7)
d = (5- -3)2 + (7-2)2
2-D: “THE” Distance formula
(-3,2)
(5,7)
d = (8)2 + (5)2
2-D: “THE” Distance formula
(-3,2)
(5,7)
d = 64 + 25
2-D: “THE” Distance formula
(-3,2)
(5,7)
d = 89 = 9.434
Who can do the Pythagorean Theorem?
(-3,2)
(5,7)d
( 5,2)
Who can do the Pythagorean Theorem?
(-3,2)
(5,7)
d = 82 + 52
( 5,2)
d
8
5
Who can do the Pythagorean Theorem?
(-3,2)
(5,7)
d = 64 + 25
( 5,2)
d
8
5
The distance formula and the Pythagorean Theorem are very similar.