A locus (plural loci) is a set of all the points that satisfy a particularly mathematical statement or rule. Such statements or rules can contain any number of distinct variables, but we can only easily graphically represent statements with two or three distinct variables.
If a statement contains only one distinct variable, the graph of the locus can be represented on a number line. For instance, here are the graphs for \(x = 3\) and \(x > -1\), respectively:
If a statement contains exactly two distinct variables, the graph of the locus is most typically represented on a coordinate plane, where one variable (typically called \(x\)) is measured by a horizontal number line and the other variable (typically called \(y\)) is measure by a vertical number line. These lines are called axes (singular axis).
Here, for instance, are the graphs for \(y=3x\):
and \(y>=2x^2-5x+3\):
If a statement contains exactly three distinct variable, the third variable is typically called \(z\), and the graph is a flattened representation of coordinate space. Here is the graph for \(z=2x^2-y^3\):
In addition to mathematical statements, a locus can be described by a rule, such as: A circle is the locus of points that are equidistant from a given point. To specify a circle, we would need the point and the distance.
These describe the same locus:
- The set of all points four units away from the point (3, 2).
- \((x-3)^2+(y-2)^2=16\)
The locus is shown here in red; the central point is in green:
In the case of parabolas, the rule statement and the functional statement involve different points. A parabola is the locus of points that are equidistant from a given point (called a focus) and a given line (called a directrix); a parabola is the graph of a quadratic function with a specific vertex and vertical stretch. The focus and the vertex are not the same point.
These describe the same locus:
- The set of points equidistant from the focus \((1,-2)\) and the line \(y=-4\).
- \(y=0.25(x-1)^2-3\)
Note that this has a vertex of \((1, -3)\). The vertex is shown in the diagram as green, the focus and directrix in purple, and the locus (a parabola) in red:
You can explore more using Desmos. The orange line segments in the graph will always be the same length, regardless of how you adjust the variables.
Defining the parabola as a locus according to a rule (rather than an equation) allows for more flexibility for positioning. In particular, the directrix does not need to be parallel to the \(x\)-axis (explore more on Desmos):
