Question

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

Answer 1

`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