Match the functions (a)-(f) with their graphs (A)-(F) in...

Match the functions (a)-(f) with their graphs (A)-(F) in Figure 23. (a) $f(x, y)=|x|+|y|$ (b) $f(x, y)=\cos (x-y)$ (c) $f(x, y)=\frac{-1}{1+9 x^{2}+y^{2}}$ (d) $f(x, y)=\cos \left(y^{2}\right) e^{-0.1\left(x^{2}+y^{2}\right)}$ (e) $f(x, y)=\frac{-1}{1+9 x^{2}+9 y^{2}}$ (f) $f(x, y)=\cos \left(x^{2}+y^{2}\right) e^{-0.1\left(x^{2}+y^{2}\right)}$

Match the functions (a)-(f) with their graphs (A)-(F) in Figure 23.
(a) $f(x, y)=|x|+|y|$
(b) $f(x, y)=\cos (x-y)$
(c) $f(x, y)=\frac{-1}{1+9 x^{2}+y^{2}}$
(d) $f(x, y)=\cos \left(y^{2}\right) e^{-0.1\left(x^{2}+y^{2}\right)}$
(e) $f(x, y)=\frac{-1}{1+9 x^{2}+9 y^{2}}$
(f) $f(x, y)=\cos \left(x^{2}+y^{2}\right) e^{-0.1\left(x^{2}+y^{2}\right)}$

Key Concepts

Graphing Functions of Two Variables

This concept involves understanding how a function of two variables defines a surface in three-dimensional space. By interpreting the output value as the height above each point (x, y) in the domain, one can visualize the topography of the surface, including features like peaks, valleys, ridges, and troughs.

Absolute Value Functions and Their Graphs

Absolute value functions, particularly when involving multiple variables, lead to surfaces that are piecewise linear. They often produce sharp edges or ridges (such as a pyramid or cone-like shape) due to the non-smoothness at the point(s) where the input changes sign, highlighting the importance of symmetry and linear segments in the surface.

Periodic Functions in Multiple Variables

Periodic functions, especially those involving trigonometric functions like cosine, result in wave-like patterns that repeat regularly over the domain. In functions of two variables, the periodicity can depend on combinations of the variables (e.g., x-y), leading to patterns such as stripes or ripples that reflect repeating oscillations.

Rational Function Graphs

Rational functions, where the function is given by the ratio of polynomials, typically exhibit behavior such as smooth depressions or peaks that gradually approach asymptotic values. Their graphs may display different scaling or stretching along various directions depending on the coefficients of the polynomial terms.

Exponential Decay and Gaussian Envelopes

Exponential decay functions, particularly those expressed with a term like e^(-constant*(x^2+y^2)), impose a damping effect on the surface. Such damping often acts as a Gaussian envelope that causes the surface features to diminish in amplitude away from the center, leading to a localized behavior with rapid decay at the periphery.

Modulated Oscillatory Behavior

Combining an oscillatory component (like a cosine function) with an exponential decay factor results in modulated oscillations. The cosine term creates a rippling or wave-like effect, while the exponential term gradually suppresses these oscillations as one moves away from the central region, producing a pattern where the amplitude of the oscillations decreases with distance.

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