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, ...

Graphs of Sine and Cosine 1.2 An applet illustrating how the graphs of sine and cosine are related to the unit circle. Transformations of Functions 1.3 An applet illustrating how transformations ...

👉 Learn how to graph quadratic equations in vertex form. A quadratic equation is an equation of the form y = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic equation is in ...

A function, 𝑓(𝑥) relates an input and an output. The equation 𝑦 = 𝑥² can be written as the function 𝑓(𝑥) = 𝑥². The notation for a function is 𝑓(𝑥) or 𝑔(𝑥) or ℎ(𝑥) and so on. Sometimes ...

👉 Learn how to graph exponential functions involving vertical shift. An exponential function is a function that increases rapidly as the value of x increases. To graph an exponential function, it is ...