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:

Three-Dimensional Cartesian Coordinate System Section 8.6a

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

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.

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

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???

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???

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

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

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

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?

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!!!

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.

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???

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