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A contour map is a graphical representation where level curves of a function are plotted in one plane. These curves represent the set of points where the function has the same constant value, often referred to as "contours". In this exercise, the different contour plots correspond to the values the function, specifically the multiplication of two variables, can take. A contour map allows for an easy visualization of how a function behaves over a range of values.

For instance, if we consider the function of two variables, such as our function defined as \(f(x, y) = xy\), the different level curves are plotted for several values of \(c\). A contour map helps to quickly identify the different regions of the function corresponding to the different level sets. By sketching these curves on one set of axes, we can see all the curves simultaneously, making it easy to understand how the function behaves across its domain.

When plotting a function like \(f(x, y) = xy = c\), the resulting shapes for \(c eq 0\) are hyperbolas. Hyperbolas are types of conic sections that appear as opposite-facing curves on a coordinate plane. They typically occur when a function of two variables forms an expression like \(xy=c\).

In our exercise, the hyperbolas that form are distinguished by whether \(c\) is positive or negative. For negative values of \(c\), hyperbolas open towards the second and fourth quadrants. For positive values of \(c\), they appear in the first and third quadrants. These hyperbolas demonstrate symmetry around the origin, and the steepness depends on the absolute value of \(c\).

It's also notable that for \(c = 0\), the level curve is simply where \(xy=0\), which corresponds to the axes themselves, meaning the x-axis and y-axis.

The coordinate axes, which consist of the x-axis and y-axis, play an essential role in the sketching of level curves. They serve as a reference frame in which the shapes, like our hyperbolas, can be drawn and understood. The axes also serve a specific function in our particular problem.

When \(c = 0\), the level curve for \(xy = c\) is coincident with the coordinate axes themselves, because the product \(xy = 0\) either needs \(x=0\) or \(y=0\). Hence, it leads to two lines on the plane — the x-axis and y-axis. This uniqueness stems from the fact that multiplying any number by zero yields zero, highlighting the importance of understanding the function in the context of its axes.

Understanding the orientation and the position of these axes helps in sketching and interpreting the curves correctly, noting where they intersect or align with the axes across the various values of \(c\).

A function of two variables is an equation that takes two input values and produces a single output value. In mathematical terms, this type of function is expressed as \(f(x, y)\). Functions of two variables have a wide array of applications in fields such as physics, engineering, economics, and more.

In this context, our function is \(f(x, y) = xy\). Such a function implies that for any pair \((x, y)\), the product \(xy\) yields a particular value. The level curve aspect of this function illustrates how different constant outputs (\(c\) values) shape the geometry of the level curves.

Working with functions of two variables requires comfort with graphical analysis because they often cannot be simplified to a graph of just one variable. Visualizing these functions means dropping down one dimension to produce contour maps, which can represent complex 3D surfaces as 2D visuals. The key aspect is to understand how these two variables interact to form these surfaces, as observed in their level curves.