Assignment 3: Investigation 2

The purpose of this assignment is to explore the distinguished characteristics of a hyperbola as well its inverse, in addition to a linear function presented to intersect the hyperbolic function. It is important to note that significant changes take place when the selected the constant c is changed. This is also true when the original equation of the Hyperbola exhibits its natural characteristics with the two curves directly reflecting each other and the graph approaching the y-axis but never touching it. The y-axis clearly serves as its asymptote.

When the constant c is negated to -1, the two curves not only shift positions, but completely change the configuration. It appears to be to inverse of the original function. The horizontal linear line with the slope of zero, intersecting the original function yields two negative roots while yielding one positive and one negative root on the inverse function.  The horizontal line will ultimately intersect the curve in the curve in the fourth quadrant, where the roots of the original function will be positive at the vertex.

Under investigation 2, the same mathematical phenomenon occurs as the original function and the second one are inverse functions of each other as the constant c is negated. In addition, the new line, 2x +b =0, with a negative slope of -2, now serves a line of reflection intersecting the original function while it does not intersect the inverse function. The line serves a line of symmetry as well on the inverse function.

As the constant c increases, the characteristic of the original hyperbolic function remains the same yet stretching the curve. As the constant c is negated, the characteristics of the inverse function also remain the same as the curve equally stretches.

In the investigation, when the constant changes to n, the hyperbolic function becomes a three dimensional figures where the lines become two dimensional planes and the original functions become three dimensional curves. One interesting discovery was when the equation is no longer expressed as an equation and written as an expression, the graph becomes three dimensional and the linear expression is a surface with an area.