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Mathematics portfolio - Translations.

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Introduction

Man Ju    Y12D

Mathematics - portfolio

Translations

1.        

image00.png

2.        image01.png is the effect of translation vector            of image02.png. It moves up 2 units.  

image14.png is the effect of translation vector            of image02.png. It moves up 4 units.

image26.png is the effect of translation vector            of image02.png. It moves down 3 units.

3.

image37.png

image02.png+ 3 is the effect of translation vector             of image02.png. It moves up 3 units.

image02.png-1 is the effect of translation vector             of image02.png. 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 image03.png. If the number is positive, the curve will be pulled upwards. If the number is negative, the curve will be pulled downwards.

5.        

image04.png

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.

...read more.

Middle

. If the number is positive, the curve will shift to the left. If the number is negative, the curve will shift to the right.

image12.png

image13.png= sin (x-90)2  is the effect of translation vector             of image13.png= 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 image13.png= 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 functionimage13.png.  

10.        i.        image15.png

                      Shift 3 to the right and 5 downwards.

ii.        image16.png

           Shift 2 to the left and 1 downwards.

        iii.        image17.png, (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.        

image18.png

2.        image19.pngis the effect of one way stretch along the y axis, scale factor 2 of image02.png.

image20.png is the effect of one way stretch along the y axis, scale factor 4 of image02.png.

image21.png

...read more.

Conclusion

image34.png

image13.png= 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 image13.png= 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 functionimage13.png.  

All Together!

10.        i.        image35.png

                      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.        

image36.png

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.        image38.png

                      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

image39.png

  1. y = x2 shift 1 to the right to become y = (x-1)2
  2. 2. y = (x-1)2 shift 5/3 down to become y = (x-1)2 –5/3
  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]

...read more.

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