Lecture 2: Ordered Pairs and the Cartesian Plane

Question 1

ordered pair

Answer 1

An ordered pair (a,b) of two real numbers is defined such as (a,b) =/ (b,a)
- the set of ordered pairs (x,y) is denoted by R2 and is called the Cartesian product of R with itself
- geometrically R2 is represented by two real number lines intersecting perpendicular

Question 2

slope

Answer 2

• given two ordered pairs (x1,y1) and (x2, y2)
  the ratio y2-y1/x2-x1 is denoted by m and is called the slope of line passing through these points

Question 3

example: (-4,0) and (2,2)

Answer 3

m= y2-y1/x2-x1 = 2-0/2-(-4) = 2/ 2+4 = 2/6 = 1/3

Question 4

two special cases

Answer 4

m = 1-1/2-3 = 0/-1 = 0
m= -3-0 / 2-2 = -3/0 = undefined

Question 5

lines in the plane

Answer 5

• horizontal lines in the plane are described by the equation y=b where b is a constant
• vertical lines in the plane are described by the equation x=a where a is a constant

Question 6

equation of lines ; 3 cases to consider

Answer 6

1. find the equation of the line passing through the points ( y-y1 = y2-y1/x2-x1[x-x1)
   y-y1 = m (x-x1) = y-0 = 2-0/2+ 4(x–4)
   y= 2/6 ( x+4) = 1/3 (x+4)
2. find the equation of a line with slope 3 that passes through the point (-4,0)
   y-y1 =m (x-x1)
   y-0 = 3 (x+4)
   y= 3x+12
3. find the equation of a line with slope 3 and y intercept 2
   y=mb+b = y= 3x + 2

Question 7

integer exponents

Answer 7

• let a be a real number and n be a positive integer
• we define a^n

Question 8

properties of exponents

Answer 8

• a^m X a^n = a^ m+n
• a^m/a^n = a^ m-n
• (a^m)^n = a^mn
• (ab)^n = a^n X B^n
• (a/b)^n = A^n/b^n
• (a/b)^-n = (b/a)^n
• a^-n/b^-m= b^m/a^n
• a^0 = 1
• a^-n = (1/a^n) = (1/a)^n