Analytic geometry is actually another name for coordinate (or Cartesian) geometry, a way to geometrically represent ordered pairs of numbers (or coordinates). The objects are described as n-tuples of points, in which n = 2 in a plane and n = 3 in a space (or two and three dimensions) in some coordinate system. Overall, analytic geometry allows mathematicians to determine the position, configurations, and separations ofobjects using algebraic representation and manipulation of equations.
In analytic geometry, a graph is simply a way of plotting—thus, visually representing— points, lines, curves, and solids in order to understand and interpret certain geometric figures and to solve equations. For example, solving an equation with one to two variables (usually written as x and y, or two dimensions) results in a curve on a graph (note: a line is considered a curve in geometry). Equations that contain three variables (usually as x, y, and z, or three dimensions) result in a surface.
A coordinate system is one that uses coordinates—a number or numbers that identify a point on a number line, plane, or in space. These points are most often seen on a graph and can be a combination of two numbers for a two-dimensional figure or three numbers for three dimensions.