Use the contours of the linear function z=f(x, y) in Figure...

Use the contours of the linear function $z=f(x, y)$ in Figure 12.77 to create possible contour labels for the linear function $z=g(x, y)$ satisfying the given condition.
(Figure cant copy)

The graph of $g$ is parallel to the graph of $f$ and passes through the point $(2,2,0).$

Key Concepts

Linear Functions

A linear function in two variables is typically expressed as z = ax + by + c. Such a function represents a plane in three-dimensional space. The coefficients a and b determine the slope or tilt of the plane, while the constant c determines its vertical position. Understanding the form and behavior of linear functions is essential in analyzing how changes in the equation affect its graph.

Contour Lines / Level Sets

Contour lines, or level sets, are curves along which a function of two variables takes on a constant value. For a linear function, these contours are straight and parallel lines in the x-y plane. They provide a two-dimensional representation of a three-dimensional surface, allowing one to visualize the gradient and rate of change of the function without needing to plot the entire surface.

Parallel Planes and Translations

When one plane is translated vertically, the resulting plane remains parallel to the original. In the context of linear functions, this involves altering the constant term while keeping the coefficients of x and y the same. This concept allows for the generation of a family of parallel planes, each differing only by the vertical shift, which directly affects the contour labels without changing their directional orientation.

Determining Translations with a Point

When a specific point is known to lie on a translated plane, that point can be used to determine the translation parameter. By substituting the coordinates of the point into the equation of the plane, one can solve for the constant term that translates the original plane such that it passes through the given point. This method is crucial for aligning parallel surfaces with prescribed conditions.

Relationship Between Contour Labels of Parallel Functions

For two parallel linear functions, the contour lines in the x-y plane remain identical in orientation and spacing. However, the numerical labels (the function values) on those contours will differ by a constant amount corresponding to the vertical translation between the two functions. This relationship is important when transferring contour information from one function to construct or analyze another with similar geometric features.

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