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3.5 Graphs in Three Dimensions 2.Graphing Equations in Three Dimensions

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2) Graphing Equations in Three Dimensions The graph of a 3D equation is a plane in 3- dimensional space. x -x y -y z -z

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2) Graphing Equations in Three Dimensions It is difficult to draw planes, so instead we draw traces. A trace is a line that passes through two intercepts. x -x y -y z -z

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2) Graphing Equations in Three Dimensions It is difficult to draw planes, so instead we draw traces. A trace is a line that passes through two intercepts. x -x y -y z -z

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2) Graphing Equations in Three Dimensions It is difficult to draw planes, so instead we draw traces. A trace is a line that passes through two intercepts. x -x y -y z -z yz-trace xy-trace xz-trace Crosses the y- and z- axes Crosses the x- and z- axes Crosses the x- and y- axes

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To draw a trace, we need to know the x, y and z intercepts. Example 1: Sketch the graph of x + y + z = 5. 2) Graphing Equations in Three Dimensions

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Example 1: Sketch the graph of x + y + z = 5. x-intercept: y = 0, z = 0 x + 0 + 0 = 5 x = 5 2) Graphing Equations in Three Dimensions

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Example 1: Sketch the graph of x + y + z = 5. y-intercept: x = 0, z = 0 0 + y + 0 = 5 y = 5 2) Graphing Equations in Three Dimensions

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Example 1: Sketch the graph of x + y + z = 5. z-intercept: x = 0, y = 0 0 + 0 + z = 5 z = 5 2) Graphing Equations in Three Dimensions

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Example 1: Sketch the graph of x + y + z = 5. x-intercept = 5 y-intercept = 5 z-intercept = 5 2) Graphing Equations in Three Dimensions

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Example 1: Sketch the graph of x + y + z = 5. x-intercept = 5 y-intercept = 5 z-intercept = 5 2) Graphing Equations in Three Dimensions x -x y -y z -z (0, 0, 5) (0, 5, 0) (5, 0, 0)

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Example 1: Sketch the graph of x + y + z = 5. x-intercept = 5 y-intercept = 5 z-intercept = 5 2) Graphing Equations in Three Dimensions x -x y -y z -z (0, 0, 5) (0, 5, 0) (5, 0, 0)

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Example 2: Sketch the graph of 2x – y + 3z = 6. 2) Graphing Equations in Three Dimensions

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Example 2: Sketch the graph of 2x – y + 3z = 6. x-int 2x – 0 + 3(0) = 6 2x = 6 x = 3 2) Graphing Equations in Three Dimensions

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Example 2: Sketch the graph of 2x – y + 3z = 6. x-int y-int 2x – 0 + 3(0) = 62(0) – y + 3(0) = 6 2x = 6 -y = 6 x = 3 y = -6 2) Graphing Equations in Three Dimensions

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Example 2: Sketch the graph of 2x – y + 3z = 6. x-int y-int z-int 2x – 0 + 3(0) = 62(0) – y + 3(0) = 6 2(0) – 0 + 3z = 6 2x = 6 -y = 6 3z = 6 x = 3 y = -6 z = 2 2) Graphing Equations in Three Dimensions

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Example 2: Sketch the graph of 2x – y + 3z = 6. x-int = 3 y-int = -6 z-int = 2 2) Graphing Equations in Three Dimensions x -x y -y z -z (0, 0, 2) (3, 0, 0) (0, -6, 0)

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Example 2: Sketch the graph of 2x – y + 3z = 6. x-int = 3 y-int = -6 z-int = 2 2) Graphing Equations in Three Dimensions x -x y -y z -z (0, 0, 2) (3, 0, 0) (0, -6, 0)

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Example 2: Sketch the graph of 2x – y + 3z = 6. x-int = 3 y-int = -6 z-int = 2 2) Graphing Equations in Three Dimensions x -x y -y z -z (0, 0, 2) (3, 0, 0) (0, -6, 0)

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Homework p.149 #21-23, 30, 31, 33, 37-40 QUIZ TOMORROW: 3.1, 3.2

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Equations in two variables can be represented by a graph. Each ordered pair (x,y) that makes the equation true is a point on the graph. Graph equation.

Equations in two variables can be represented by a graph. Each ordered pair (x,y) that makes the equation true is a point on the graph. Graph equation.

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