How to do this question regarding matrices and transformation (quadratic/cubic)?

enter image source here

1 Answer
Feb 14, 2018

#color(blue)(y=-1/9x^3-12)#

Explanation:

All points on the line # y=x^3+4# will be transformed by the matrix

#bb(A)=((3,0),(0,-3))#

Any point on the line will have coordinates of the form.

#(k,k^3+4)#

Thus:

#((x'),(y'))=((3,0),(0,-3))((k),(k^3+4))#

#color(white)(888888)=((3k+0),(0-3(k^3+4)))=((3k),(-3k^3-12))#

.i.e.

#x'=3k#

#y'=-3k^3-12#

Eliminating #k#:

#x=3k=>k=x/3#

#y=-3(x/3)^3-12#

#color(blue)(y=-1/9x^3-12)#

We can check this by generating an #x# and a #y# value from the original equation, putting these through the transformation and testing the results in our equation of the image curve:

#x=1# and #y=5#

#((3,0),(0,-3))((1),(5))=((3),(-15))#

Plugging in #x# in:

#y=-1/9x^3-12#

#y=-1/9(3)^3-12=-15color(white)(88)# as expected