1. 3.5 Graphs in Three Dimensions (Day 1)

2. Terms
Coordinate Space: added a third axis, the z-axis, to the xy-coordinate plane
Ordered Triples: (x, y, z)

3. z axis
y axis
x axis

4. (x, y)
(2, 4)

5. (2, 3, 4) x y z + - + - + - (x, y, z) Out, back Right, left Up, down
+ -
+ -
+ -
(2, 3, 4) Up
Out
Right

6. Why is the x-axis slanted?
(x, y, z)
z axis
(2, 3, 4)
y axis
x axis
Why is the x-axis slanted?

7. Graphing in Coordinate Space
Graph each point in coordinate space.
a. (-3, 3, -4) b. (5, 0, 2)

8. White Boards Graph each point in coordinate space.
1. (4, -5, 6) (-7, 3, -2)
Describe the location of each point in coordinate space.
3. (-3, -2, 8) (6, 4, -5)
back 3, left 2, up 8
out 6, right 4, down 5

9. 3.5 Graphs in Three Dimensions (Day 2)

10. yz – trace xz – trace xy – trace Trace -
The line formed when a plane intersects one of the coordinate planes. z
yz – trace
(no x dimension)
xz – trace
(no y dimension)
x = 0
y = 0
front board
windows y
xy – trace
(no z dimension) x
z = 0
floor

11. x – ( , , ) y – ( , , ) z – ( , , ) 6 2 3 xy – trace: z = 0 x + 3y = 6
Graph each equation and find the equation of each trace.
Everything is building upon what we’ve already learned… the graph is now a plane instead of a line.
Intercepts:
x – ( , , )
y – ( , , )
z – ( , , ) 6
x = 6 2
0 + 3y + 0 = 6 3
z = 6
Traces:
xy – trace:
z = 0
x + 3y = 6
xz – trace:
y = 0
x + 2z = 6
yz – trace:
x = 0
3y + 2z = 6

12. Keep in mind…
x: (4, 0)
y: (0, 3)
Different From…
(4, 3)

13. x – ( , , )
y – ( , , )
z – ( , , ) 6 2 3
Different From…
(6, 2, 3)

14. White Boards 1. 9x – 3y + z = 9 2. – 4x + 2y – 6z = 12
Graph each equation.
1. 9x – 3y + z = – 4x + 2y – 6z = 12
9x = 9
– 4x = 12
– 3y = 9
2y = 12
z = 9
– 6z = 12

15. White Boards 3. – 4x + 3y – 6z = – 24 4. 5x + 10y – 4z = 20
Graph each equation and find the equation of each trace.
3. – 4x + 3y – 6z = – x + 10y – 4z = 20
– 4x = – 24
5x = 20
3y = – 24
10y = 20
– 6z = – 24
– 4z = 20
– 4x + 3y = – 24
5x + 10y = 20
– 4x – 6z = – 24
5x – 4z = 20
3y – 6z = – 24
10y – 4z = 20