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2 Dimensional graphs 3 Dimensional graphs Functions and graphs Graphing functions

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Quit -5-4-3-2-112345 x -5 -4 - 3 - 2 - 1 1 2 3 4 5 y y = x y = x 2 y = x 3 y = x 4

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Quit x 2 + y 2 = 9 CircleEllipse x 2 + = 1 9 __ y2y2

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Quit Three dimensions + + = 1 4 __ y2y2 4 x2x2 4 z2z2 Sphere

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Quit Three dimensions x 2 + + z 2 = 1 9 __ y2y2 Cigar shape

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Quit Three dimensions Flying saucer + y 2 + = 1 9 __ z2z2 9 x2x2

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Quit -5-4-3-2-112345 x -5 -4 - 3 - 2 - 1 1 2 3 4 5 y y = x + 1 By observation a lot can be deduced about a graph, if both powers of x and y are one, it is always a straight line

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Quit -5-4-3-2-112345 x -5 -4 - 3 - 2 - 1 1 2 3 4 5 y y + 1 = x 2 If one power of x or y is one and the other is two it’s a curve with one bend

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Quit -5-4-3-2-112345 x -5 -4 - 3 - 2 - 1 1 2 3 4 5 y y = x 3 – 3x 2 – x + 3 If one power of x or y is one and the other is three it’s a curve with two bends

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Quit -5-4-3-2-112345 x If one power of x or y is one and the other is four it’s a curve with three bends -5 -4 - 3 - 2 - 1 1 2 3 4 5 y y = x 4 – 5x 3 + 5x 2 + 4x – 4

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Quit Draw the graph of the curve y = 2x + 3 from x = – 3 to x = 3 x = – 3 y = 2(– 3) + 3 = – 6 + 3 = – 3(– 3, – 3) x = – 2 y = 2(– 2) + 3 = – 4 + 3 = – 1(– 2, – 1) x = – 1 y= 2(– 1) + 3 = – 2 + 3 = 1(– 1, 1) x = 0 y= 2(0) + 3 = 0 + 3 = 3(0, 3) y = 2x + 3

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Quit Draw the graph of the curve y = 2x + 3 from x = – 3 to x = 3 x = 1 y = 2(1) + 3 = 2 + 3 = 5(1, 5) x = 2 y = 2( 2) + 3 = 4 + 3 = 7( 2, 7) x = 3 y= 2(3) + 3 = 6 + 3 = 9 (3, 9) y = 2x + 3

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Quit -3-2123 x -4 -3 -2 y 1 2 3 4 5 6 7 8 9 10 y = 2x + 3 (– 3, – 3) (– 2, – 1) (– 1, 1) (1, 5) (2, 7) (3, 9) (0, 3)

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Quit Draw the graph of the curve y = x 3 – 3x 2 – 4x + 12 from x = – 3 to x = 3 x = – 3 y = (– 3) 3 – 3(– 3) 2 – 4(– 3) + 12 = – 27 – 27 + 12 + 12 = – 30(– 3, – 30) x = – 2 y = (– 2) 3 – 3(– 2) 2 – 4(– 2) + 12 = – 8 – 12 + 8 + 12 = 0(– 2, 0) x = – 1 y= (– 1) 3 – 3(– 1) 2 – 4(– 1) + 12 = – 1 – 3 + 4 + 12 = 12(– 1, 12) x = 0 y= (0) 3 – 3(0) 2 – 4(0) + 12 = 0 – 0 – 0 + 12 = 12(0, 12) y = x 3 – 3x 2 – 4x + 12

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Quit Draw the graph of the curve y = x 3 – 3x 2 – 4x + 12 from x = – 3 to x = 3 x = 1 y = (1) 3 – 3(1) 2 – 4(1) + 12 = 1 – 3 – 4 + 12 = 6(1, 6) x = 2 y = (2) 3 – 3( 2) 2 – 4( 2) + 12 = 8 – 12 + 8 + 12 = 0( 2, 0) x = 3 y= (3) 3 – 3(3) 2 – 4(3) + 12 = 27 – 27 – 12 + 12 = 0(3, 0) y = x 3 – 3x 2 – 4x + 12

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Quit -3-21234 x -4 -2 2 4 6 8 10 12 14 y y = x 3 – 3x 2 – 4x + 12 (– 1, 12) (0, 12) (3, 0) (– 2, 0) (2, 0) (1, 6)

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Quit Draw the graph of the curve x 2 + y 2 = 9 (0) 2 + y 2 = 9 x 2 + y 2 = 9 y 2 = 9 x = 0 y = 3 (0, 3)(0, – 3) x 2 + (0) 2 = 9 x 2 = 9 y = 0 x = 3 (3, 0)(– 3, 0)

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Quit x 2 + y 2 = 9 Circle r = 3 (0, 3)(0, – 3) (3, 0) (– 3, 0)

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Quit Draw the graph of the curve y = x 4 – 5x 3 + 5x 2 + 5x – 6 from x = – 2 to x = 3 x = – 2 y = (– 2) 4 – 5(– 2) 3 + 5(– 2) 2 + 5(– 2) – 6 = 16 + 40 + 20 – 10 – 6 = 0(– 2, 60) x = – 1 y= (– 1) 4 – 5(– 1) 3 + 5(– 1) 2 + 5(– 1) – 6 = 1 + 5 + 5 – 5 – 6 = 0(– 1, 0) x = 0 y= (0) 4 – 5(0) 3 + 5(0) 2 + 5(0) – 6 = 0 – 0 + 0 + 0 – 6 = – 6(0, – 6) y = x 4 – 5x 3 + 5x 2 + 5x – 6

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Quit Draw the graph of the curve y = x 4 – 5x 3 + 5x 2 + 5x – 6 from x = – 2 to x = 3 x = 1 y = ( 1) 4 – 5(1) 3 + 5( 1) 2 + 5(1) – 6 = 1 – 5 + 5 – 10 – 6 = 0(1, 0) x = 2 y= (2) 4 – 5(2) 3 + 5(2) 2 + 5(2) – 6 = 16 – 40 + 20 + 10 – 6 = 0(2, 0) x = 3 y= (3) 4 – 5(3) 3 + 5(3) 2 + 5(3) – 6 = 81 – 135 + 45 + 15 – 6 = 0(3, 0) y = x 4 – 5x 3 + 5x 2 + 5x – 6

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Quit -2 x 1234 -7 -6 -5 -4 -3 -2 1 2 3 4 5 y (0, – 6) (3, 0)(– 1, 0)(2, 0) (1, 0) y = x 4 – 5x 3 + 5x 2 + 5x – 6

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Quit y 2 = 9 x = 0 y = 3 (0, 3)(0, – 3) x 2 = 4 y = 0 x = 2 (2, 0)(– 2, 0) + = 1 9 __ y2y2 4 x 2 Draw the graph of the curve + = 1 9 __ y2y2 4 x 2 = 9 __ y2y2 1= 4 x2x2 1

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Quit -212 x -3 -2 1 2 3 y Ellipse + = 1 9 __ y2y2 4 x2 x2 (0, – 3) (– 2, 0) (2, 0) (0, 3)

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