Graphing Surfaces in R3

GROUP MEMBERS:

1. _____________________________

2. ______________________________

3. ______________________________

4. ______________________________

 

 

Goal: To develop an understanding of three dimensional surfaces gotten as the graph of a function z = f(x,y). We will use the method of slices and both make sketches and use manipulatives.

  1. z = f(x,y) = x2 + y2

In lecture, we have discussed how to make "slice" the surface. We set y=constant and then get z as a function of x. We then set x = constant and get z as a function of y. We take several different values for x=constant, each time getting a "slice". We then put these slices together.

Instructions: Each person in the team will have a pipe cleaner that they will bend into the appropriate shape for a slice. You will build up the surface by putting the slices (i.e. pipe cleaners) together. Each person read your instructions out loud and then carry out your task. Your teammates can assist by helping to hold the surface together.

Person 1: You have the y=0 slice which gives z = f(x,0) = x2. Make this slice out of blue. Align this slice above the x axis.

Person 2: You have the x =0 slice which gives z = f(0, y) = y2. Make this slice out of red. Attach this to the first slice. It should be above the y axis.

Person 3: You have the x=1 slice which gives z = f(1, y) = 1 + y2. Make this slice out of red. Attach this to the first slice. It should be parallel to the y axis.

Person 4: You have the x=-1 slice which gives z = f(1, y) = 1 + y2. Make this slice out of red. Attach this to the first slice. It should be parallel to the y axis.

Person 1: You have the y=1 slice which gives z = f(x, 1) = x2 + 1. Make this slice out of blue. Attach this to the x=0 slice at the appropriate place. It should be parallel to the x axis.

Also connect it to the x = 1 slice and the x=-1 slices.

 

Person 2: Help Person 1 out by calculating the points of intersection:

when y=1 and x =0 then z =

when y=1 and x =1 then z =

when y =1 and x =-1 then z =

Person 3: You have the y=-1 slice which gives z = f(x, 1) = x2 + 1. Make this slice out of blue. Attach this to the x=0 slice at the appropriate place. It should be parallel to the x axis.

Also connect it to the x = 1 slice and the x = -1 slices.

 

Person 2: Help Person 1 out by calculating the points of intersection:

when y= -1 and x =0 then z =

when y= -1 and x =1 then z =

when y = -1 and x =-1 then z =

 

 

 

 

 

2. z = f(x,y) = x2 - y2

First everyone will calculate some slices. Then the group will make the pipe cleaner model. Finally, each person will try to make their own complete sketch putting all the slices together.

 

Each person do these calculations on your own piece of paper. Their should be 4 sheets attached to this packet. Each person take your own sheet and do your work on the sheet. When you have done part (a) and (b), then return to discuss your results with the group and record your answers on the next page.

Building the surface z = x2 - y 2

  1. Recording data: Fill in the following table.

    2. x = 0, z = f(0, y) =

    x = 2, z = f(2, y) =

    x = -2, z = f(-2, y) =

    y = 0, z = f(x, 0) =

    y= 2, z = f(x, 2) =

    y = - 2, z = f(x, -2) =

  2. Use pipe cleaners to build the surface. Use the usual color scheme: x = constant slices out of red, y = constant slices out of blue.

    3. Person 1: the y = 0 slice

    Person 2: the x = 0 slice

    Person 3: the x = 2 slice

    Person 4: the x = - 2 slice

    Person 2: the y = -2 slice

    Person 1: the y = 2 slice.

     

     

    z = f(x,y) = x2 - y2

    Person 2 : a) Set y = 0. Sketch z = f(x,0) = . (A slice along the x direction).

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    b) Set x = 2. Sketch z = f(2,y) = . (A slice along the y direction).

     

     

     

    c) When the model is done, try sketching the complete surface using either

  3. the y=0 backbone and the x=0, x=2, x=-2 ribs or
  4. the x=0 backbone and the y=0, y=2, y=-2 ribs.

    5.  

     

     

     

     

    z = f(x,y) = x2 - y2

    Person 3 : a) Set x = 0. Sketch z = f(0,y) = . (A slice along the y direction).

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    b) Set y = 2. Sketch z = f(x,2) = . (A slice along the x direction).

     

     

     

    c) When the model is done, try sketching the complete surface using either

  5. the y=0 backbone and the x=0, x=2, x=-2 ribs or
  6. the x=0 backbone and the y=0, y=2, y=-2 ribs.

    7.  

     

     

     

    z = f(x,y) = x2 - y2

    Person 4 : a) Set y = 0. Sketch z = f(x,0) = . (A slice along the x direction).

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    b) Set x = -2. Sketch z = f(-2,y) = . (A slice along the y direction).

     

     

     

    c) When the model is done, try sketching the complete surface using either

  7. the y=0 backbone and the x=0, x=2, x=-2 ribs or
  8. the x=0 backbone and the y=0, y=2, y=-2 ribs.

 

 

 

 

 

z = f(x,y) = x2 - y2

Person 1 : a) Set x = 0. Sketch z = f(0,y) = . (A slice along the y direction).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b) Set y = -2. Sketch z = f(x,-2) = . (A slice along the x direction).

 

 

 

c) When the model is done, try sketching the complete surface using

  1. (Person 2 and Person 4) the y=0 backbone and the x=0, x=2, x=-2 ribs or
  2. (Person 3 and Person 1) the x=0 backbone and the y=0, y=2, y=-2 ribs.