Any general direction on visualizing functions in 3 dimensions?

Question

I have a problem like this:

In which I have to match the graph to the function.

What's the best way to approach a problem like this? Certainly, I could just graph it with software and see what comes out, but is there a methodology that I can apply here? I know some are obvious like the absolute value function is probably the middle-right one with only positive values and looks like a V except in 3 dimensions. But how would I visualize the others or other ones like them in the future? What things should I look for?

A link to a video or any other similar resource would be fine. I'm just looking for ideas.

Thank you very much.

Answer 1

I believe there is no "best" way to approach this, but you can use some basic understanding of elementary functions and common sense to identify which graph corresponds to which function.

For example, the graph of $|x|+|y|$ must be the right-hand-sided in the second row. The others are too smooth. For the second and third functions in your function-column, the only candidates are the the left-hand-sided graphs in the first and third row, because only those graphs lie strictly in the region $z<0$. Looking at the function $-\frac{1}{1+9x^2+9y^2}$ you observe that there is symmetry under rotations, which is only displayed by the left-hand-sided graph in the first row. We are left with three functions containing $\cos $. You can see that $\cos(x-y)$ is constant along the lines $y-x=c$, and the graph displaying this property is left-hand-side, second row. Finally, rotational symmetry helps to distinguish between the last two. Can you tell which is which?

Answer 2

The best you can do is take or $X$ or $Y$ as a constant and see the "nivel curve" of the graph. For example, if you take $z$ as a constant $k$ for $z=f(x,y) = |x|+|y|$, you will have that every plane $XY$ that intersects the graph should have the figure of $$k = |x|+|y| ~~ (\text{a square})$$ and then look for graphs where any plane XY with variable $z$ has the form of square.

If you take $x$ as constant, you should look at plane $YZ$. If you take $y$ as constant, you should look at plane $XZ$.

What varibale take as constant? The one which makes easier the result equation.