Graphs of Equations

In two-dimensional analytic geometry, the graph of an equation in the variables x and y is a curve in R².

In three-dimensional analytic geometry, the graph of an equation in the variables x, y and z is a surface in R³.

Example 1 - describe a surface in R³

Describe the surface in R³ represented by each equation.

(a) z = 3 (b) y = 5

Solution

(a). In R³ , the equation z = 3 represents the set of all points

{ (x, y, z) | x, y ε R, z = 3 }

This set contains infinitely many points.

However, each point (x, y, 3) in this set has a common property: each point lies 3 units above the xy-plane.

That is, this set of points is the horizontal plane that is parallel to the xy-plane and is 3 units above it.

(b) Similarly, in R³ , the equation y = 5 represents the set of all points

{ (x, y, z) | x, z ε R, y = 5 }

Each point (x, 5, z) in this set lies 5 units above the xz-plane.

So this set of points is the vertical plane that is parallel to the xz-plane and is 5 units to its right.

Let k be a constant (a real number). Then

(1) The graph of the equation x = k in R³ is a plane parallel to the yz-plane;

(2) The graph of the equation y = k in R³ is a plane parallel to the xz-plane;

(3) The graph of the equation z = k in R³ is a plane parallel to the xy-plane;

YOU TRY IT

Describe the region of R³ that is represented by the inequality y > 5.