Exploring the Graph of y (-1)^x: A Comprehensive Analysis

The function y (-1)^x exhibits fascinating behavior both in terms of its real and complex representations. This article explores the graph of this function, examining its distinct patterns and the underlying mathematical principles that govern its behavior.

Behavior of the Function

The function ( y (-1)^x ) behaves differently depending on whether ( x ) is an integer or non-integer. Specifically, for integer values of ( x ), the function can be described as follows:

If ( x ) is an even integer (e.g., 0, 2, 4, ...), then ( y 1 ). If ( x ) is an odd integer (e.g., 1, 3, 5, ...), then ( y -1 ).

This results in a series of discrete points at ((n, 1)) and ((n, -1)) for each integer ( n ).

Graph Description: Discrete Nature and Vertical Lines

The graph of ( y (-1)^x ) consists of isolated points at ( y 1 ) for even integers and ( y -1 ) for odd integers, with no values defined for non-integer ( x ).

Discrete Nature: The graph exhibits a discrete pattern, with each point corresponding to an integer value of ( x ). There are no points between these integers because ( y (-1)^x ) is not defined for non-integer values of ( x ). Vertical Line: When sketched, the graph appears as two horizontal lines: one at ( y 1 ) for even integers and one at ( y -1 ) for odd integers. These lines are connected by gaps, representing the undefined points for non-integer ( x ).

Three-Dimensional Representation

Intriguingly, the graph of ( y (-1)^x ) can also be represented in three dimensions by plotting the complex-valued function. By Euler's formula, we can express ( (-1)^x ) as ( e^{ipi x} ), where ( e ) is the base of the natural logarithm, and ( i ) is the imaginary unit.

Parametric Representation

Using Euler's formula, we can parametrize the curve in three dimensions as ( (x, cos(pi x), -sin(pi x)) ). This transformation allows us to visualize the function in a more comprehensive way.

The following graph is rendered using Geogebra, with the ( x )-axis, ( y )-axis, and ( z )-axis represented by red, green, and blue respectively:

Intersection with the Plane z0

The intersection of this three-dimensional curve with the plane ( z 0 ) (which corresponds to the imaginary part of ( y ) being zero) yields the standard graph of ( y (-1)^x ) for integer values of ( x ). This graph alternates between 1 and -1 for consecutive values of ( x ).

Variants of this technology include plotting ( y a^x ) for real numbers ( a ), plotting the complex roots of polynomials with real coefficients such as ( y x^{21} ), and exploring a wide range of functions in both real and complex domains.

Understanding the behavior of ( y (-1)^x ) provides insights into the interplay between real and complex numbers, and serves as a fascinating case study in the realm of mathematical functions and their graphical representations.

Related Keywords

Keywords: y (-1)^x, graph, complex numbers, real numbers, Euler's formula, parametric representation, three-dimensional graph, complex roots, polynomial functions.