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 $g$ has the same contour as $f$ for the value $z=30$ but is different from the graph of $f.$

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 $g$ has the same contour as $f$ for the value $z=30$ but is different from the graph of $f.$

Key Concepts

Linear Functions

Linear functions in two variables take the form z = ax + by + c. Their graphs are planes in three-dimensional space, and their level curves, or contours, are straight lines in the xy?plane. Because the relationship is linear, any changes in the function such as scaling or translation will result in a corresponding shift or rescaling in both the graph and the contour map.

Level Curves (Contours)

Level curves, or contours, of a function are the sets of points in the domain where the function takes on a constant value. For a linear function, these curves are typically straight lines. Contour maps help visualize how the function’s value changes across the domain, and specific contours (such as the one at z = 30 mentioned) can be used as reference points when comparing different functions.

Function Transformation and Scaling

Transformations of functions, including scaling and translation, can alter the graph of a function while preserving certain features such as a particular level curve. In the context of the given problem, even though g is different from f overall, they share the same contour at z = 30. This indicates a transformation where the relative positions of level curves change except for that specific level, illustrating how function parameters can shift contour labels while retaining some level set invariants.

Contour Labeling

Contour labeling involves assigning specific constant values to the level curves of a function. When comparing two functions that share one contour value, the labels on the remaining contours provide information about how one function has been transformed relative to the other. The task of creating contour labels for g based on f emphasizes understanding how changes in the function’s coefficients affect each contour level.

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