is the effect of translation vector of . It moves right 3 units.
is the effect of translation vector of . It moves left 1 unit.
is the effect of translation vector of . It moves left 4 units.
8.
y = (x + 2)2 is the effect of translation vector of . It moves left 2 units.
y = (x - 1)2 is the effect of translation vector of . It moves right 1 unit.
9. Curves move either left or right horizontally. The units they move is according to the number after x in the equation . If the number is positive, the curve will shift to the left. If the number is negative, the curve will shift to the right.
= 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 .
is the effect of one way stretch along the y axis, scale factor 1/2 of .
is the effect of one way stretch along the y axis, scale factor -3 of .
3.
4. Curves are being stretched one way along the y axis, the scale factor are according to the number in front of x2 in the equation . If the number is positive, the curve will be stretched in the positive y value region. But if the number is negative, the curve will be stretched up side down to the negative y value region. If the number is smaller than 1, the curve will be stretched outwards. If the number is bigger than 1, the curve will be stetched inwards.
5.
= 2sin(x) is the effect of one way stretch along the y axis, scale factor 2 of = sinx. This has the same effect with the previous examples. Scale factor is determined by the number in front of sinx in the equation = a sin (x). If the number is positive, the curve will be stretched in the positive y value region. But if the number is negative, the curve will be stretched up side down to the negative y value region. If the number is smaller than 1, the curve will be stretched outwards. If the number is bigger than 1, the curve will be stretched inwards. Therefore the generalization extend to any function.
6.
7. is the effect of one way stretch along the x axis, scale factor 1/2 of .
is the effect of one way stretch along the x axis, scale factor 2 of .
is the effect of one way stretch along the x axis, scale factor 1/3 of .
is the effect of one way stretch along the x axis, scale factor 3 of .
8.
y = (-2x)2 is the effect of one way stretch along the x axis, scale factor 1/2 of .
y = (-1/2x)2 is the effect of one way stretch along the x axis, scale factor 2 of .
9. Curves are being stretched one way along the x axis, the scale factor are one over the number in front of x2 after it has been squared in the equation . But the number is in the square bracket, that means the number is being squared. So the number is always positive(signs can be ignored). 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.
= 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]
