Shielded Metal Arc Welding SMAW Vertical Down and Vertical Up Welding of Pipelines
Vertical shifts (up) A familiar example: Vertical shift up 3: y-values each increase by 3 graph is...
13
Vertical shifts (up) x -5 -4 -3 -2 -1 0 1 2 3 4 5 y 25 16 9 4 1 0 1 4 9 16 25 A familiar example: 2 gx x x -5 -4 -3 -2 -1 0 1 2 3 4 5 y 28 19 12 7 4 3 4 7 12 19 28 Vertical shift up 3: 2 3 f x x y-values each increase by 3 graph is shifted up 3 units -10 -5 0 5 10 15 20 -8 -6 -4 -2 0 2 4 6 8 2 gx x -10 -5 0 5 10 15 20 -8 -6 -4 -2 0 2 4 6 8 2 3 f x x
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Transcript of Vertical shifts (up) A familiar example: Vertical shift up 3: y-values each increase by 3 graph is...
Vertical shifts (up)
x -5 -4 -3 -2 -1 0 1 2 3 4 5
y 25 16 9 4 1 0 1 4 9 16 25
A familiar example: 2g x x
x -5 -4 -3 -2 -1 0 1 2 3 4 5
y 28 19 12 7 4 3 4 7 12 19 28
Vertical shift up 3: 2 3f x x
y-values each increase by 3
graph is shifted up 3 units-10
-5
0
5
10
15
20
-8 -6 -4 -2 0 2 4 6 8
2g x x
-10
-5
0
5
10
15
20
-8 -6 -4 -2 0 2 4 6 8
2 3f x x
x -5 -4 -3 -2 -1 0 1 2 3 4 5
y 20 11 4 -1 -4 -5 -4 -1 4 11 20
2 5f x x Vertical shift down 5:
y-values each decrease by 5
x -5 -4 -3 -2 -1 0 1 2 3 4 5
y 25 16 9 4 1 0 1 4 9 16 25
Original curve: 2g x x
graph is shifted down 5 units
More vertical shifts (down)
-10
-5
0
5
10
15
20
-8 -6 -4 -2 0 2 4 6 8
2g x x
-10
-5
0
5
10
15
20
-8 -6 -4 -2 0 2 4 6 8
2 5f x x
Horizontal shifts (right)
x -5 -4 -3 -2 -1 0 1 2 3 4 5
y 64 49 36 25 16 9 4 1 0 1 4
x -5 -4 -3 -2 -1 0 1 2 3 4 5
y 25 16 9 4 1 0 1 4 9 16 25
Original curve: 2g x x
23f x x Horizontal shift right 3:
y-values are shifted to the right 3 units
graph is shifted right 3 units-10
-5
0
5
10
15
20
-8 -6 -4 -2 0 2 4 6 8
-10
-5
0
5
10
15
20
-8 -6 -4 -2 0 2 4 6 8
2g x x 2
3f x x
x -5 -4 -3 -2 -1 0 1 2 3 4 5
y 1 0 1 4 9 16 25 36 49 64 81
x -5 -4 -3 -2 -1 0 1 2 3 4 5
y 25 16 9 4 1 0 1 4 9 16 25
Original curve: 2g x x
More horizontal shifts (left)
24f x x Horizontal shift Left 4:
y-values are shifted to the left 4 units
graph is shifted left 4 units-10
-5
0
5
10
15
20
-8 -6 -4 -2 0 2 4 6 8
2g x x
-10
-5
0
5
10
15
20
-8 -6 -4 -2 0 2 4 6 8
24f x x
Summary of vertical and horizontal shifts
f x g x k
Given a function g whose graph is known, and a positive
number k, the graph of the function f is:
graph of g, shifted up k units
f x g x k
f x g x k
f x g x k
graph of g, shifted down k units
graph of g, shifted right k units
graph of g, shifted left k units
CAUTION:the signs here may be counter-intuituve!
Reflections about the x-axis
x -5 -4 -3 -2 -1 0 1 2 3 4 5
y und und und und und 0.0 1.0 1.4 1.7 2.0 2.2
x -5 -4 -3 -2 -1 0 1 2 3 4 5
y und und und und und 0.0 -1.0 -1.4 -1.7 -2.0 -2.2
undefined if x < 0
Original curve:
Reflected about x-axis: f x x
-1.0
0.0
1.0
2.0
3.0
4.0
-6 -4 -2 0 2 4 6
-4.0
-3.0
-2.0
-1.0
0.0
1.0
-6 -4 -2 0 2 4 6
g x x
g x x
f x x
y-values each replaced by their opposite
x -5 -4 -3 -2 -1 0 1 2 3 4 5
y 2.2 2.0 1.7 1.4 1.0 0.0 und und und und und
undefined if x > 0
Reflections about the y-axis
x -5 -4 -3 -2 -1 0 1 2 3 4 5
y und und und und und 0.0 1.0 1.4 1.7 2.0 2.2
undefined if x < 0
Original curve:
Reflected about y-axis: f x x
-1.0
0.0
1.0
2.0
3.0
4.0
-6 -4 -2 0 2 4 6
g x x
-1.0
0.0
1.0
2.0
3.0
4.0
-6 -4 -2 0 2 4 6
g x x f x x
mirror image of y-values
domain: 0,
domain: ,0
Summary of reflections
f x g x
Given a function g whose graph is known, the graph of the
function f is:
graph of g, reflected about the x-axis
f x g x graph of g, reflected about the y-axis
domain of f is "opposite" of domain of g
i.e. if domain of g is [a,b] then domain of f is [-b,-a]
domain of f is domain of g
Vertical stretching
g x
A cubic polynomial:
-10
-8
-6
-4
-2
0
2
4
6
8
10
-8 -6 -4 -2 0 2 4 6 8
x -5 -4 -3 -2 -1 0 1 2 3 4 5y -20 -1.6 7.2 8.8 5.6 0 -5.6 -8.8 -7.2 1.6 20 2f x g x
g x
f x
each y-value doubles
x -5 -4 -3 -2 -1 0 1 2 3 4 5y -10 -0.8 3.6 4.4 2.8 0 -2.8 -4.4 -3.6 0.8 10
Vertical Shrinking
g x
The same cubic:
1
3f x g x
x -5 -4 -3 -2 -1 0 1 2 3 4 5y -3.3 -0.3 1.2 1.5 0.9 0 -0.9 -1.5 -1.2 0.3 3.3
x -5 -4 -3 -2 -1 0 1 2 3 4 5y -10 -0.8 3.6 4.4 2.8 0 -2.8 -4.4 -3.6 0.8 10
each y-value shrinks by 1/3
-10
-8
-6
-4
-2
0
2
4
6
8
10
-8 -6 -4 -2 0 2 4 6 8
g x
f x
Horizontal Stretching
1
2f x g x
g x
The same cubic:
Y-values are stretched out from the centerx -5 -4 -3 -2 -1 0 1 2 3 4 5
y 4.4 4.4 3.8 2.8 1.5 0.0 -1.5 -2.8 -3.8 -4.4 -4.4
x -5 -4 -3 -2 -1 0 1 2 3 4 5y -10 -0.8 3.6 4.4 2.8 0 -2.8 -4.4 -3.6 0.8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-8 -6 -4 -2 0 2 4 6 8
g x
f x
Horizontal shrinking
2f x g x
The same cubic
x -5 -4 -3 -2 -1 0 1 2 3 4 5y -10 -0.8 3.6 4.4 2.8 0 -2.8 -4.4 -3.6 0.8 10
x -5 -4 -3 -2 -1 0 1 2 3 4 5y -170 -78 -25 -0.8 4.4 0.0 -4.4 0.8 25 78 170
g x
-10
-8
-6
-4
-2
0
2
4
6
8
10
-8 -6 -4 -2 0 2 4 6 8
g x
f x
Summary of stretching and shrinking
f x cg x
Given a function g whose graph is known, and a positive
number c, the graph of the function f is:
graph of g, stretched vertically
f x g cx
graph of g, shrunk vertically
graph of g, shrunk horizontally
graph of g, stretched horizontally
1c
1c
1c
1c
