Solution: The dilation of gk by a scale factor of 2 centered at (-2,-4) is [tex]g'(2,4)[/tex] and [tex]k'(10,12)[/tex].
Explanation:
It is given that the scale factor is 2 and center of dilation is (-2,-4).
If the center of dilation is (a,b) with scale factor k, then the dilation of a point P(x,y) is determined by the given formula.
[tex]D(x,y)=(a+k(x-a),b+k(y-b))[/tex] .....(1)
To find the g' substitute x=0, y=0, a=-2, b=-4 and k=2 in equation (1).
[tex]g'=(-2+2(2),-4+2(4))\\g'=(2,4)[/tex]
To find the k' substitute x=4, y=4, a=-2, b=-4 and k=2 in equation (1).
[tex]k'=(-2+2(4+2),-4+2(4+4))\\g'=(10,12)[/tex]
Therefore, the dilation of gk by a scale factor of 2 centered at (-2,-4) is [tex]g'(2,4)[/tex] and [tex]k'(10,12)[/tex].
Coordinates of g(0,0) and k(4,4).
If we have to dilate gk by a scale factor of 2 , it means we have to multiply each coordinate by 2.
g(0×2,0×2) =p(0,0)⇒ New coordinate
k(4×2,4×2)=q(8,8)⇒ New Coordinate
This is the procedure when we dilate a shape when center is Origin.
Now ,as you have given center is not origin.
It's center is (-2,-4).
So, we will use the following Formula.
If a point(x,y) and is dilated by a factor k and it's center is (m,n) then
New position=[m+k(x-m),n+k(y-n)]
So,applying the formula ,we get new coordinates of g,k
g(0,0)=g'[-2+2{0-(-2)},-4+2{0-(-4)}]=g'(2,4)
k(4,4)=k'[-2+2{4-(-2)},-4+2{4-(-4)}]=(10,12)
