What does $f(x,y) = x + y$ mean in graphing?

Question

All right, so I'm supposed to graph these $5$ things: $$x \geq 3\\ x\leq 6\\ y \geq 3\\ y \leq 6\\ f(x,y) = x + y$$

I was able to graph the first four, but I have no idea what the last one means. I searched it up on Google and I found 3-D images, so I'm absolutely clueless on what to do here.

Answer 1

So, my understanding is that you need help understanding the intuition of what a function of two variables looks like graphically. My advice is to consider the limiting behavior in a few directions.

Consider the following cases. What if we follow the path $y=x$ on in the $xy-$plane? We can see that as $(x,y)\to\infty$, $f(x,y)\to\infty$. So, based on the situation, we should imagine that the function increases linearly, because $y=x\implies f(x,x)=2x=2y$. What about along the $x$ and $y$ axes?

Along the $x-$axis, we can see that our function is $f(x,0)=x$. So, we should imagine the function increasing linearly, with half the rate of change of along $x=y$. Likewise for $f(0,y)=y$. Now consider the line $y=-x$. We can see that along this line, $f(x,-x)=0$, and so the line orthogonal to $y=x$ is $0$. This should paint the picture of a plane in your head, but just in case, maybe we should look at some points.

$f(1,1)=2$, $f(1,0)=1$, $f(0,1)=1$, $f(0,0)=0$, $f(1,-1)=0$, $f(-1,1)=0$, $f(-1,-1)=-2.$ Plotting these on a $3-$dimensional grid should give a clear idea of the image: a plane. Here's an image.

Answer 2

You are correct in assuming that it is three-dimensional. For the sake of simplicity, let's replace $f(x,y)$ with $z$. So now we can say $z = x+y$. So, the bigger $x$ gets, the bigger $z$ gets, and the bigger $y$ gets, the bigger $z$ gets. Now draw three-dimensional axes (if you've ever done any drafting or AutoCAD, just imagine the isometric view), and call the vertical axis $z$, the horizontal axis $y$, and the diagonal axis $x$. If both $x$ and $y$ are very large, $z$ is very large. If both $x$ and $y$ are very small, $z$ is very small. Now thing about when $x$ is negative and $y$ is positive, or vice versa. Can you visualize what will happen to $z$? When is $z=0$?