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How do the coordinate axes divide a plane into four quadrants?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

In analytic geometry, the Cartesian plane is defined by two perpendicular axes—the horizontal x-axis and the vertical y-axis—that intersect at a point called the origin. These axes divide the plane into four distinct regions known as quadrants, which are typically labeled I through IV based on the signs of the coordinates $(x, y)$ within them:

Quadrant I: Both $x$ and $y$ values are positive.
Quadrant II: The $x$ value is negative, and the $y$ value is positive.
Quadrant III: Both $x$ and $y$ values are negative.
Quadrant IV: The $x$ value is positive, and the $y$ value is negative.

This system allows for the precise algebraic description of points and geometric shapes within a two-dimensional space.1