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When it comes to understanding the basics of algebra, linear functions are a fundamental concept. A linear function describes a constant rate of change and is graphically represented by a straight line in 2D space. The general form of a linear function in two variables is given by the equation \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept.

In a 3D context, a linear function involving two independent variables, such as \( f(x, y) \) from the exercise, extends this idea to describe a plane. The plane is determined by how the function affects each of the variables. Algebraically, this is represented as \( z = ax + by + c \), where \( z \) depends linearly on \( x \) and \( y \) and \( a \) and \( b \) represent the inclination of the plane with respect to the x-axis and y-axis, respectively, while \( c \) is the function's value when both \( x \) and \( y \) are zero, also known as the z-intercept.

Plotting graphs in three-dimensional space may seem daunting, but it follows the same principles as 2D graph plotting, just with an additional axis. For the 3D graph, each point is represented by an ordered triple \( (x, y, z) \), where \( x \) and \( y \) are the horizontal and vertical positions in the plane, and \( z \) is the height or depth.

When plotting the graph of a 3D function, such as \( f(x, y) \) from our example, being able to locate intercepts and understand the orientation of planes is essential. Visualization tools or 3D graphing software can be very helpful for this purpose. For instance, graphing the function \( f(x, y) = 1 - x - y \) will reveal a plane cutting through the 3D coordinate system. Recognizing how the function behaves with increasing or decreasing values of \( x \) and \( y \) aids in understanding the shape and position of this geometric figure in space.

The points where a graph crosses the axes are known as axis intercepts. They are fundamental to graph sketching as they provide key coordinates for the construction of the graph. In 2D, there are just x and y-intercepts, but in 3D plotting, we also have z-intercepts, which indicate where the graph crosses the z-axis.

In the case of our function \( f(x, y) \), we determine these intercepts by setting two variables to zero and solving for the remaining variable. This results in points such as \( (1, 0, 0) \) for the x-intercept, \( (0, 1, 0) \) for the y-intercept, and \( (0, 0, 1) \) for the z-intercept. These intercepts are crucial for accurately sketching and understanding the dimensions and location of the graph in the 3D coordinate system.

In a 3D graph, plane inclination refers to the angle a plane makes with the horizontal plane or the axes within the coordinate system. It determines the slope or steepness of the plane. For a linear function like \( f(x, y) \), the inclination can be understood in terms of the coefficients of \( x \) and \( y \) in the equation.

The coefficients \( -1 \) for both \( x \) and \( y \) in the function \( f(x, y) = 1 - x - y \) suggest that the plane has the same 'downward' slope in both the x and y directions. The negative sign indicates that the function's value decreases when \( x \) or \( y \) increase. When sketching the graph, the plane will thus appear to slant away from the viewer, descending equally in both directions from the axis intercepts.