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 A point is named by an

ordered pair

written as Quadrant II Origin: (0, 0) Quadrant I (x, y) → (3, 2) (x, y) → (-2, -1) x - axis Quadrant III Quadrant IV

Distance Formula (pp. A27, A28)

d

 ((

x

2 

x

1 ) 2  (

y

2 

y

1 ) 2 ) • Comes from the Pythagorean Theorem • An error just means you typed it in wrong

The Midpoint Formula (pp. A28-A29)

M

   

x

1 

x

2 2 ,

y

1  2

y

2    • Think of it as averaging 2 coordinates • Your answer is always an ordered pair

Equation of a Circle (pp. A29-A30) (

x



h

) 2  (

y



k

) 2 

r

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, x axis, 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)