# Functions of Two

You can tell which axis is x and which is y by the right hand convention. If you push the positive x axis into the positive y axis by curling the fingers of your right hand, your thumb points in the direction of positive z.

# Variables

## Coordinate systems

First a digression to make sure that we all understand graphing in three dimensions. (For now, we will focus on rectangular coordinates.)

Where before we located points in a plane by determining distances from the origin along two perpindicular lines, we now do the same thing -- and more. First, imagine rotating the x-y plane so that it moves from being perpendicular to your gaze to being "flat" with your gaze. Then think of that plane being located in space.

We locate a point in space by first locating a point in the x-y plane (as before) that is right "under" (or "over") the point in question, then we go some distance above or below the x-y plane to reach it. That distance is the point's z coordinate.

Play (double-click) the movie below to see how points are located. In this movie, each point on the unit circle with coordinates (x,y) is elevated to a height of z = 1 - xy.

Finally, we generate surfaces by determining a value for z at every point (x,y) in the plane.

## Covariation

We will extend the idea of covariation of two quantities values to the idea of the covaration of three quantitie's values. Recall that in the idea of function as covariation we thought of one variable's value varying and another quantity's value being determined by some rule that depended on it.

The extension of that idea is this: Suppose we have a third variable whose value depends on the values of two other quantities. The key to imagining the behavior of such functions graphically is to manage smartly how we envision these variations happening.

As mentioned above, we generate a surface by finding a value of the third variable for each point (x,y) in the plane and plotting a point that far perpendicularly from the point (x,y). There are many creative ways to envision covariation of three variables. The key is to think of generating the x-y plane efficiently. One way of thinking that is particularly powerful is to think of sweeping a curve through a plane and evaluating the function at every point of each curve, keeping track of the curves in space generated thereby. Two cases are

Each of these methods is discussed.

## Method 1: Sweep the x-y plane by moving a line perpendicular to an x- or y- axis.

Recall the assignment in which you explained the behaviors of y = x2 + nx and of y = x3 + nx so that one explanation captured what was going on in both cases.

### Make sense of y = x2 + nx (in a new way)

Task 1. Play (double-click) the movie below to see the variation of y = x2 + nx as n varies from -2 to 2. After looking at it, play it again with the idea that you are looking straight down an "n" axis, from negative (near you) to positive (on the other side of the visible coordinate system). Do this until you can visualize it.

Task 2. After succeeding on Task 1, play (double-click) the movie below to see the variation of y = x2 + nx as n varies from -2 to 2. This movie, however, shows the "n" axis! Play it and the first movie until you can see that they are the same movie from different perspectives. (The movie on the right also shows the line perpendicular to the "n" axis whose points are the axis on which the value of x varies for each value of n.)

Task 3. After succeeding on the Task 2, imagine that the curve in Task 2 is covered with fairy dust. The fairy dust will generate a surface composed of points. You should be able to describe these points and what they represent!. Play (double-click) the move below to see how these curves generate a surface.

## Make sense of y = x3 + nx (in a new way)

Task 3. Play (double-click) the movie below to see the variation of y = x3 + nx as n varies from -2 to 2. After looking at it, play it again with the idea that you are looking straight down an "n" axis, from negative (near you) to positive (on the other side of the visible coordinate system). Do this until you can visualize it.

Task 4. After succeeding on Task 3, play (double-click) the movie below to see the variation of y = x3 + nx as n varies from -2 to 2. This movie, however, shows the "n" axis! Play it and the previous movie until you can see that they are the same movie from different perspectives. (The movie on the right also shows the line perpendicular to the "n" axis whose points are the axis on which the value of x varies for each value of n.)

Task 6. After succeeding on the Task 5, imagine that the curve in Task 5 is covered with fairy dust. The fairy dust will generate a surface composed of points. You should be able to describe these points and what they represent!. Play (double-click) the move below to see how these curves generate a surface.

The idea of level curves along the z-axis is helpful when the formula that defines the function generates "nice" curves as solutions to equations in two variables.

For example, consider the function z = x2 + 3y2. We know that each value of z determines an ellipse.

• When z is 1, it determines the ellipse generated by 1 = x2 + 3y2. This ellipse is elevated to a height of 1 above the x-y plane (because z=1 for each x-y pair that satisfies the equation).
• When z is 1.5, it determines the ellipse generated by 1.5 = x2 + 3y2. This ellipse is elevated to a height of 1.5 above the x-y plane (because z=1.5 for each x-y pair that satisfies the equation).
• When z is 1.75, it determines the ellipse generated by 1.75 = x2 + 3y2. This ellipse is elevated to a height of 1.75 above the x-y plane (because z=1.75 for each x-y pair that satisfies the equation).
• and so on.

Play (double-click) the movie below to see ellipses generated by n = x2 + 3y2 for n between 0 and 7.

Play (double-click) the movie below to see the above pattern played out for z ranging between 0 and 7. In watching the movie, you need to see things happening in this order:

• A value for z is set (call it c). A plane is graphed at the elevation of c.
• A solution set is determined for z = c. Graph that solution set in the x-y plane. (This solution set is called the pre-image of c -- the set of points that have c as their image when mapped onto z.)
• Project that solution set to the elevation of c. This elevated projection is called a level curve for z = c.

Play the movie repeatedly until you can see these steps happening repeatedly and automatically as z varies from 0 to 7.

Now, imagine the level curves being made of fairy dust. They will sweep out a surface. Play the movie above again until you can imagine the surface that the level curves generate. Then play (double-click) the movie below to check yourself.

NOTE: Even though it seems as if the z axis is where the action is, the initial work is done in the x-y plane. It is in the x-y plane that we find a solution set for each value of z, and then we project it to a height of (that value of) z. The x-y plane gets "swept" because every point that plays a role in generating the surface is identified because (conceptually) we "find" the pre-image of each value of z.