# 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]

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

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