Post on 01-Apr-2015
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2 Dimensional graphs
3 Dimensional graphs
Functions and graphs
Graphing functions
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-5 -4 -3 -2 -1 1 2 3 4 5 x
-5
-4
-3
-2
-1
1
2
3
4
5 y
y = x
y = x2
y = x3
y = x4
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-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
x2 + y2 = 9
Circle Ellipse
x2 + = 19__y
2
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Three dimensionsThree dimensions
+ + = 14__y
2
4__x
2
4__z
2
Sphere
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Three dimensionsThree dimensions
x2 + + z2 = 19__y
2
Cigar shape
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Three dimensionsThree dimensions
Flying saucer + y2 + = 1 9__z
2
9__x
2
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-5 -4 -3 -2 -1 1 2 3 4 5 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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-5 -4 -3 -2 -1 1 2 3 4 5 x
-5
-4
-3
-2
-1
1
2
3
4
5 y
y + 1 = x2
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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-5 -4 -3 -2 -1 1 2 3 4 5 x
-5
-4
-3
-2
-1
1
2
3
4
5 y
y = x3 – 3x2 – 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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-5 -4 -3 -2 -1 1 2 3 4 5 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 = x4 – 5x3 + 5x2 + 4x – 4
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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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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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-3 -2 -1 1 2 3 x
-4
-3
-2
-1
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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Draw the graph of the curve y = x3 – 3x2 – 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 = x3 – 3x2 – 4x + 12
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Draw the graph of the curve y = x3 – 3x2 – 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 = x3 – 3x2 – 4x + 12
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-3 -2 -1 1 2 3 4x
-4
-2
2
4
6
8
10
12
14
y
y = x3 – 3x2 – 4x + 12
(– 1, 12) (0, 12)
(3, 0)(– 2, 0) (2, 0)
(1, 6)
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Draw the graph of the curve x2 + y2 = 9
(0)2 + y2 = 9
x2 + y2 = 9
y2 = 9x = 0 y = 3
(0, 3) (0, – 3)
x2 + (0)2 = 9 x2 = 9y = 0 x = 3
(3, 0) (– 3, 0)
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-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
x2 + y2 = 9
Circle r = 3
(0, 3)(0, – 3)
(3, 0)
(– 3, 0)
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Draw the graph of the curve y = x4 – 5x3 + 5x2 + 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 = x4 – 5x3 + 5x2 + 5x – 6
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Draw the graph of the curve y = x4 – 5x3 + 5x2 + 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 = x4 – 5x3 + 5x2 + 5x – 6
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-2 -1x
1 2 3 4
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
y
(0, – 6)
(3, 0)(– 1, 0) (2, 0)(1, 0)
y = x4 – 5x3 + 5x2 + 5x – 6
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y2 = 9x = 0 y = 3
(0, 3) (0, – 3)
x2 = 4y = 0 x = 2
(2, 0) (– 2, 0)
+ = 19
__y2
4__ x2
Draw the graph of the curve + = 19
__y2
4__ x2
=9__y
2
1
=4__x
2
1
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-2 -1 1 2x
-3
-2
-1
1
2
3
y
Ellipse + = 19__y
2
4__ x2
(0, – 3)
(– 2, 0) (2, 0)
(0, 3)
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