Semigroups of transformations and endomorphisms have emerged as powerful algebraic frameworks to elucidate the underlying structures of graphs. By harnessing the principles of semigroup theory, ...

A translation is a movement of the graph either horizontally parallel to the \(x\)-axis or vertically parallel to the \(y\)-axis. The graph of \(f(x) = x^2\) is the same as the graph of \(y = x^2\).

Draw the line of reflection. From a point on the shape, count how far it is vertically from the line of reflection. Count the same amount horizontally on the other side of the line and plot the point.

Math fans can now spin, zoom, and explore complex equations in full 3D right on their iPad with iPadOS 26. Here's what it looks like. With iPadOS 26, Apple is introducing interactive 3D graphing in ...