Lesson B.1
The Cartesian Plane
Objective
Students will:
Plot points on the Cartesian plane and sketch scatter plots
Use the distance formula to determine the distance between two points
Use the Midpoint Formula to find the midpoint between two points
Find the equation of a circle
Translate points in the plane
The Cartesian Plane (pp. A25, A26)
• Know the parts
– x-axis, y-axis, origin, quadrants, coordinates
• Be able to graph a point
• Sketch a scatter plot
– Independent variable on x-axis (time)
The Coordinate Plane
y - axis
Quadrant II
Origin:
(0, 0)
Quadrant I
A point is
named by an
ordered pair
written as
(x, y) → (3, 2)
(x, y) → (-2, -1)
x - axis
Quadrant III
Quadrant IV
Distance Formula (pp. A27, A28)
d  (( x2  x1 )  ( y2  y1 ) )
2
2
• Comes from the Pythagorean Theorem
• An error just means you typed it in wrong
The Midpoint Formula (pp. A28-A29)
 x1  x2 y1  y2 
M 
,

2 
 2
• Think of it as averaging 2 coordinates
• Your answer is always an ordered pair
Equation of a Circle (pp. A29-A30)
( x  h)  ( y  k )  r
2
•
•
•
•
2
2
Center of the circle is at (h, k)
If the center is (-1, -5) then equation is (x + 1)2 + (y + 5)2…
r is the radius
The distance from the center to any point on the circle is
the radius
Translating in the Plane (p. A31)
• Adding values to an ordered pair will shift
it in the plane
– (x + 3, y – 2) will shift three right, two down
Reflections (p. A31)
What happens when you reflect about the y-axis, xaxis, or origin?
Reflections: opposite coordinates
(x, y) → (-x, y) reflect about y-axis
(x, y) → (x, -y) reflect about x-axis
(x, y) → (-x, -y) reflect about origin
Examples
1) Find the distance between and midpoint of points (-2, 3)
and (4, -5)
2) A circle has a center at (3, 4). The point (8, 16) is on the
circle. Write an equation for this circle in standard form.
3) Give the coordinates of the following vertices of a triangle
after a reflection about the origin: (2, 3), (5, -1), (8, 6)