1. Let’s start with a little problem…
Use the fact that k = 2p is twice the focal length and half the
focal width to determine a Cartesian equation of the parabola
whose polar equation is given.
The graph???
, so…
Vertex:
And since the parabola opens left, the equation is:

2. Three-Dimensional Cartesian Coordinate System
Section 8.6a

3. z y x Drawing z = constant Practice: (0, 0, z) (0, y, z) P(x, y, z)
(x, 0, z)
(0, y, 0) y
y = constant
(x, 0, 0)
(x, y, 0) x
x = constant

4. Important Features of the 3-D Cartesian Coordinate System
Coordinate Axes – the axes labeled x, y, and z – they form the
right-handed coordinate frame.
Cartesian Coordinates of P – the real numbers x, y, and z that
make up an ordered triple (x, y, z), and locate point P in space.
Coordinate Planes – the xy-plane, the xz-plane, and the
yz-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 positive
coordinates.

5. Guided Practice
Draw a sketch that shows each of the following points.

6. Equation of a Sphere First, remind me of the definition of a circle:
Circle: the set of all points in a plane that lie a fixed distance
from a fixed point.
And the definition of a sphere?
Sphere: the set of all points that lie a fixed distance from a
fixed point.
fixed distance = radius
fixed point = center
Now, do you recall the standard equation of a circle???

7. Equation of a Sphere
A point P (x, y, z) is on a sphere with center (h, k, l )
and radius r if and only if
Quick Example: Write the equation for the sphere with its center
at (–8, –2, 1) and radius 4 3.
How do we graph this sphere???

8. New Equations But first, remind me…
Distance formula in the 2-D Cartesian Coordinate System?
Midpoint formula in the 2-D Cartesian Coordinate System?

9. Distance Formula (Cartesian Space)
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

10. Midpoint Formula (Cartesian Space)
The midpoint M of the line segment PQ with endpoints
P(x , y , z ) and Q(x , y , z ) is 1 1 1 2 2 2

11. A Quick Example Find the distance between the points P(–2, 3, 1)
and Q(4, –1, 5), and find the midpoint of the line
segment PQ.
Can we verify these answers with a graph?

12. Planes and Other Surfaces
We have already learned that every line in the Cartesian
plane can be written as a first-degree (linear) equation in two
variables; every line can be written as
How about every first-degree equation in three
variables???
They all represent planes in Cartesian space!!!

13. Planes and Other Surfaces
Equation for a Plane in Cartesian Space
Every plane can be written as
where A, B, and C are not all zero. Conversely, every
first-degree equation in three variables represents a
plane in Cartesian space.

14. Guided Practice Now where’s the graph??? Sketch the graph of
Because this is a first-degree equation, its graph is a plane!
Three points determine a plane  to find them:
Divide both sides by 60:
It’s now easy to see that the following points are on the plane:
Now where’s the graph???

15. Guided Practice
Sketch a graph of the given equation. Label all intercepts.