3-5 Graphs in Three Dimensions Objective: To graph points and equations in three dimensions

Objectives Graphing Points in Three Dimensions Graphing Equations in Three Dimensions

Vocabulary You’ve learned to graph on an xy-coordinate plane using ordered pairs. Adding a third axis, the z-axis, to the xy-coordinate plane creates coordinate space. In coordinate space, you graph points using ordered triples of the form (x, y, z)

Graphing a Coordinate Space Graph each point in the coordinate space. a. (–3, 3, –4) Sketch the axes. b. (–3, –4, 2) Sketch the axes. From the origin, move back 3 units, From the origin, move back 3 units, right 3 units and down 4 units. and up 2 units. left 4 units

Real World Example In the diagram, the origin is at the center of a cube that has edges 6 units long. The x-, y-, and z-axes are perpendicular to the faces of the cube. Give the coordinates of the corners of the cube. G(–3, 3, –3), H(–3, –3, –3) F(3, 3, –3), D(3, –3, 3), A(–3, –3, 3), B(–3, 3, 3), C(3, 3, 3), E(3, –3, –3),

Sketching a Plane Sketch the graph of –3x – 2y + z = 6. Step 1:  Find the intercepts. –3x – 2y + z = 6 –3x – 2(0) + (0) = 6 To find the x-intercept, substitute 0 for y and z. –3x = 6 x = –2 The x-intercept is –2. –3(0) – 2y + (0) = 6 To find the y-intercept, substitute 0 for x and z. –2y = 6 y = –3 The y-intercept is –3. –3(0) – 2(0) + z = 6 To find the z-intercept, substitute 0 for x and y. z = 6 The z-intercept is 6.

Continued Step 2: Graph the intercepts. Step 3: Draw the traces. Shade the plane. Each point on the plane represents a solution to –3x – 2y + z = 6.

Homework Pg 149 # 5,6,13,14,15,16,17,18,19,20