Mathematics portfolio - Translations.
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Introduction
Man Ju Y12D
Mathematics - portfolio
Translations
1.
2. is the effect of translation vector of
. It moves up 2 units.
is the effect of translation vector of
. It moves up 4 units.
is the effect of translation vector of
. It moves down 3 units.
3.
+ 3 is the effect of translation vector of
. It moves up 3 units.
-1 is the effect of translation vector of
. It moves down 1 unit.
4. Curves move either up or down vertically. The units they move is according to the number after x2 in the equation . If the number is positive, the curve will be pulled upwards. If the number is negative, the curve will be pulled downwards.
5.
f(x) = sinx –2 is the effect of translation vector of f(x) = sinx. It moves down 2 units.
This has the same effect with the previous examples. The unit it moves is according to the
number after sinx in the equation f(x) = sinx c. If the number is positive, the curve will be pulled upwards.
Middle
= sin (x-90)2 is the effect of translation vector of
= sinx. It moves right 90 units. This has the same effect with the previous examples. The units the curve moves are according to the number after x in the equation
= sin(x c). If the number is positive, the curve will shift to the left. If the number is negative, the curve will shift to the right. Therefore the generalization extend to any function
.
10. i.
Shift 3 to the right and 5 downwards.
ii.
Shift 2 to the left and 1 downwards.
iii. , (first complete the square).
y = ( x – 2 )2 – 4 + 3
y = ( x – 2 )2 – 1
Shift 2 to the right and 1 downwards.
Stretches
1.
2. is the effect of one way stretch along the y axis, scale factor 2 of
.
is the effect of one way stretch along the y axis, scale factor 4 of
.
Conclusion
= sin(2x)is the effect of one way stretch along the x axis sxale factor ½. This has the same effect with the previous examples. Scale factor is determined by the number in front of x in the equation
= sin( a sinx). If the number is bigger than 1 after it has been squared, the curve will be stretched inwards. If the number is smaller than 1 after it has been squared, the curve will be stretched outwards. Therefore the generalization extend to any function
.
All Together!
10. i.
y = 2 (x2 + 2x –1/2)
y = 2 [ (x+1)2 - 1 - 1/2 ]
y = 2 [ (x+1)2 – 3/2 ]
Therefore a = 2
p = 1
q = -3/2
ii.
1. y = x2 shift 1 to the left to become y = (x+1)2
2. y = (x+1)2 shift 3/2 down to become y = (x+1)2 –3/2
3. y = (x+1)2 –3/2 stretched one way, scale factor 2 along the y axis to become
y = 2[(x+1)2 –3/2]
iii.
y = 3 (x2 - 2x – 2/3)
y = 3 [ (x -1)2 - 1 – 2/3 ]
y = 3 [ (x -1)2 – 5/3 ]
Therefore a = 3
p = -1
q = -5/3
- y = x2 shift 1 to the right to become y = (x-1)2
- 2. y = (x-1)2 shift 5/3 down to become y = (x-1)2 –5/3
- y = (x-1)2 –5/3 stretched one way, scale factor 3 along the y axis to become
y = 3[(x-1)2 –5/3]
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