3.5 Graphs in Three Dimensions (Day 1)

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

z axis y axis x axis

(x, y) (2, 4)

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

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

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

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

3.5 Graphs in Three Dimensions (Day 2)

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

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 + 0 + 0 = 6 2 0 + 3y + 0 = 6 3 0 + 0 + 2z = 6 Traces: xy – trace: z = 0 x + 3y = 6 xz – trace: y = 0 x + 2z = 6 yz – trace: x = 0 3y + 2z = 6

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

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

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

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 = – 24 4. 5x + 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