Match the function with its graph (labeled I-VI). Give reasons for your choices. (a) f(x, y)=|x|

Match the function with its graph (labeled I-VI). Give...

Key Concepts

Functions of Two Variables

This concept involves studying functions where the output depends on two independent variables. The graph of such a function is a surface in three-dimensional space, and understanding its shape and features can be accomplished by analyzing cross?sections, contours, or level sets. Recognizing how each term in the function influences the overall geometry is a key skill in multivariable calculus and analytic geometry.

Absolute Value Functions

Absolute value functions involve the modulus operator, which returns the non-negative value of a real number. In two variables, expressions such as |x| and |y| produce surfaces that are symmetric about the coordinate axes and often yield piecewise linear components or distinctive features like sharp edges or corners. Mastering absolute value functions is crucial for understanding functions that have directional uniformity and for matching their graphs by identifying inherent symmetries.

Level Sets and Contour Analysis

Level sets or contours of a function are curves along which the function maintains a constant value. They are vital in visualizing and interpreting the topography of surfaces defined by functions of two variables. By examining the shape and symmetry of these level sets (for example, circles, ellipses, diamonds, or hyperbolas), one can infer characteristics about the original function, such as the presence of peaks, valleys, or ridges, which helps in identifying the correct graph among options.

Symmetry in Multivariable Functions

Symmetry plays a central role in analyzing functions of two variables, as many functions exhibit invariance under transformations such as reflections across the x-axis, y-axis, or lines like y = x. Recognizing symmetry helps simplify the process of graphing and understanding the function by reducing the complexity of the domain that needs to be studied. It also aids in matching functions with their graphs by comparing expected symmetric features with observed ones.

Polynomial Functions and Squaring Effects

Polynomials in two variables, particularly those involving squares or higher even powers, often produce surfaces that are non-negative and exhibit smooth, parabolic geometries. Squaring functions such as (x - y)^2 or (x^2 - y^2)^2 results in elimination of sign differences and can generate ridges, valleys, or saddle-like configurations. Understanding how squaring and other polynomial operations affect surface shape is critical for predicting graph behavior.

Trigonometric Functions and Periodicity

Trigonometric functions, especially when composed with absolute values or other modifications, introduce periodic behavior into the graph of a function. For example, a sine function applied to an expression like |x| + |y| creates wave-like oscillations that repeat over a specific interval. Recognizing these periodic patterns and how they interact with the overall domain is essential for assigning the correct graph to such a function.

Transcript

00:01 So we're asked to match the following functions with the graphs of their figures.

00:08 So starting with a, we have the absolute value of x plus absolute value of y.

00:15 First, we know that this function will always be positive.

00:19 So i'll just sketch some things we know about it.

00:23 This is positive always.

00:24 We know that it should be increasing as x and y increases.

00:32 That's an and, not a plus.

00:35 So as x and y are getting bigger, the function value is getting bigger.

00:39 So as we get further away from the center, essentially, we should be increasing.

00:45 And we also know that this should be symmetrical about the x and y axis.

00:57 So taking all these things into account, it seems like the best figure for this is figure six.

01:05 Part b.

01:09 Again, we have absolute value, so our function is always positive.

01:19 We have the absolute value of x and y multiplied together.

01:25 So again, this is going to be symmetrical about the x and y axis.

01:36 The function value should also be equal to zero when either x equals zero or y equals 0.

01:52 And we also should have something similar to a parabola.

02:01 Because if we assume that x and y are equal to each other, we end up having z equals x squared, which equals y squared.

02:17 So this is a parabola kind of type function.

02:20 So taking all those things into account, that's enough to see that figure or five represents this one.

02:29 Next part c we have one over one plus x squared plus y squared.

02:35 In order to check if it's symmetric about the x or y axis, if you substitute x for negative x or y for negative y, the function value isn't altered at all.

02:50 So again, this is symmetric about the x and y axis we can see that as x and y increases our function value decreases so the function decreases as x or y goes to infinity and whenever we're at the point zero zero we can see that f of x y would be equal to one and so from all these things we can see that the first figure makes the most sense here...

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