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