Let’s start with a little problem…Use the fact that k = 2p is twice the focal length and half thefocal width to determine a Cartesian equation of the parabolawhose polar equation is given.

12

3 3cosr

4

1 cos

1, 4e k

The graph???

2k p , so… 2,4 8p p

Vertex: , 2,0h k And since the parabola opens left, the equation is:

2 8 2y x

Three-Dimensional Three-Dimensional Cartesian Cartesian

Coordinate SystemCoordinate SystemSection 8.6aSection 8.6a

x

y

z

x = constant

y = constant

z = constant

(x, 0, 0)

(x, 0, z)

(0, 0, z) (0, y, z)

(0, y, 0)

(x, y, 0)

P(x, y, z)

DrawingPractice:

Important Features of the 3-D Cartesian Coordinate System

Coordinate Axes – the axes labeled x, y, and z – they form theright-handed coordinate frame.

Cartesian Coordinates of P – the real numbers x, y, and z thatmake up an ordered triple (x, y, z), and locate point P in space.

Coordinate Planes – the xy-plane, the xz-plane, and theyz-plane have equations z = 0, y = 0, and x = 0, respectively.

Origin – the point (0, 0, 0) where the coordinate planes meet.

Octants – the eight regions defined by the coordinate planes.The first octant contains all points in space with three positivecoordinates.

Guided PracticeGuided PracticeDraw a sketch that shows each of the following points.

2,3,5 4, 2,1 3,6, 5

Equation of a SphereEquation of a SphereFirst, remind me of the definition of a circle:

Circle: the set of all points in a plane that lie a fixed distancefrom a fixed point.

And the definition of a sphere?

Sphere: the set of all points that lie a fixed distance from afixed point.

Now, do you recall the standard equation of a circle???

2 2 2x h y k r

fixed distance = radius fixed point = center

Equation of a SphereEquation of a SphereA point P (x, y, z) is on a sphere with center (h, k, l )and radius r if and only if

2 2 2 2x h y k z l r Quick Example: Write the equation for the sphere with its centerat (–8, –2, 1) and radius 4 3.

2 2 28 2 1 48x y z

How do we graph this sphere???

New EquationsNew Equations

2 2

1 2 1 2d x x y y

But first, remind me…

Distance formula in the 2-D Cartesian Coordinate System?

Midpoint formula in the 2-D Cartesian Coordinate System?

1 2 1 2,2 2

x x y yM

Distance FormulaDistance Formula(Cartesian Space)(Cartesian Space)

2 2 2

1 2 1 2 1 2,d P Q x x y y z z

The distance d(P, Q) between the points P(x , y , z )and Q(x , y , z ) in space is

1 1 1

2 2 2

Midpoint FormulaMidpoint Formula(Cartesian Space)(Cartesian Space)

1 2 1 2 1 2, ,2 2 2

x x y y z zM

The midpoint M of the line segment PQ with endpointsP(x , y , z ) and Q(x , y , z ) is1 1 1 2 2 2

A Quick ExampleA Quick Example

, 2 17d P Q

Find the distance between the points P(–2, 3, 1)and Q(4, –1, 5), and find the midpoint of the linesegment PQ.

1,1,3M Can we verify these answers with a graph?Can we verify these answers with a graph?

Planes and Other Planes and Other SurfacesSurfaces

0Ax By C

We have already learned that every line in the Cartesianplane can be written as a first-degree (linear) equation in twovariables; every line can be written as

How about every first-degree equation in three variables???

They all represent planes in Cartesian space!!!

Planes and Other Planes and Other SurfacesSurfaces

0Ax By Cz D

Equation for a Plane in Cartesian SpaceEquation for a Plane in Cartesian Space

where A, B, and C are not all zero. Conversely, everyfirst-degree equation in three variables represents aplane in Cartesian space.

Every plane can be written as

Guided PracticeGuided Practice12 15 20 60x y z Sketch the graph of

Because this is a first-degree equation, its graph is a plane!

Three points determine a plane to find them:

15 4 3

x y z Divide both sides by 60:

5,0,0It’s now easy to see that the following points are on the plane:

0,4,0 0,0,3Now where’s the graph???Now where’s the graph???

Guided PracticeGuided PracticeSketch a graph of the given equation. Label all intercepts.

2 6y z 3x