answered
2014-07-09 23:07:55 -0600
There are two kinds of questions that he can ask about this:
- Given a function, match it to a picture
- Given a function, describe what the image might look like
The first one is akin to #7 on our worksheet where we analyze the equations given to use to try to match that to one of the function's pictures. One way to go about solving these would be to try to break up the functions that are given into the x-z and y-z versions and try to superpose an image by combining the two.
An example of this would be the function on the worksheet $f(x,y) = y^2 - x$. We know this function to be very similar to the saddle shaped function $f(x,y) = x^2-y^2$, but instead we reversed the order around the minus sign and reduced the power of $x$ by one. Matching the function to the image on the the back should be quite easy now. Because the function is set up like the saddle function, we know there will be asymptotes, a linear function, and a hyperbolic function giving us the graph in the bottom right (I think).
The second possibility is that he doesn't give any images, but instead asks us to discern a possible shape of the function. To work this sort of problem it will be very important to be able to break down the function into 2D versions of the functions as well as estimate what effect using different functions (think sine, cosine, exponential, etc.) will have on the curve.
Let's use problem #1 as this example. Breaking it down into the two 2D versions we have parabolas in the x-z plane and hyperbolas in the y-z plane. Now, how is this going to help us conjure up an image of the actual function without the help of a computer or calculator?
We must superpose now superpose these two parts onto each other. Noting that they cross each other (the pictures are technically perpendicular to each other), i'll start with the hyperbola. It looks a bit like an S shape coming down from the negative $y$ space in the positive $z$ space to the positive $y$ space in negative $z$ space.
Given any value of $y$, we can now solve where the parabolas will be situated in the x-z plane, and by combining the two images, the image of the function will come out looking like a slide. This is due to the parabolas being translated through space (think y-z plane) inn a hyperbolic fashion to create the slide.
Hope this helps
