Quadric SurfacesGraphing in 3 Dimensions

Lesson 10.2

A Point in 3D

• Points graphed with respect to three axes

• (3,2 6)

z y

x

Graphing a Plane in 3D

• Given2x - 3y + 4z = 12

• Determine intercepts• z intercept when x, y = 0• y intercept when x, z = 0• x intercept when y, z = 0

z y

x

3

-4

6

Graphing a Plane in 3D

• Consider the equation x + y = 7• z can take on any value

• Results in plane …• through the line x + y = 7

on the x-y plane• parallel to z axis

z y

x

Equation of a Sphere

• Given(x – a)2 + (y – b)2 + (z – c)2 = r2

• Center at (a, b, c)• radius = r

z y

x

•a

b

•c

Locate This Sphere

• Given

• Complete the square

2 2 2

2 2 2

4 2 8 0

4 2 8

x y z x z

x x y z z

2 2 2( 2) ( 1) 13x y z

Graphing in 3D

• Solve for z

• Change calculator mode to 3D graphing

• Enter equation in x and y into the Z= screen

2 2

2 2

( 1) 13 ( 2)

1 13 ( 2)

z x y

z x y

2 2 2( 2) ( 1) 13x y z

Graphing in 3D

• Equation gives hemisphere• note only one 3D graph at a time

• Calculator takes some time to graph all the lines of the wire frame model

• Image can be rotated• hold down cursor wheel for

continuous rotation

Graphing in 3D

• Note variety of settings for graphing window• viewing angle• number of grid lines

(more lines, longertime to graph)

• Use F1-Format to turnaxes on/off

Cylinders

• Consider a curve on a plane (here, x-y plane)• line through the curve,

not on the plane• generates a “wall” by

tracing the line along the curve

• Result is a cylinder

z y

x

Cylinders

• Curve can be• parabola• hyperbola• any function

z y

x

Quadric Surfaces

• Consider possible versions of

• Sphere• Ellipsoid• Hyperbolic paraboloid• Hyperboloid (one or two sheets)• Elliptic paraboloid

• See pg 683

2 2 2 0Ax By Cz Dxy Exz Fyz Gx Hy Iz J

Assignment

• Lesson 10.2

• Page 684

• Exercises 1, 7, 9, 13-22, 25, 33, 43