We want to convert a cubic equation of the form `y=ax^3+bx^2+cx+d` into the form `y=a(x-h)^3+k` .

(1) Lets expand the vertex form:

`y=a(x-h)^3+k`

`=a[x^3-3x^2h+3xh^2-h^3]+k`

`=ax^3-3ahx^2+3ah^2x-ah^3+k`

`=ax^3+(-3ah)x^2+(3ah^2)x+(k-ah^3)`

(2) Equate the corresponding coefficients with the equation in standard form, thus:

`a=a`

`b=-3ah`

`c=3ah^2`

`d=k-ah^3`

(3)** Then the required conversions are given by:**

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We want to convert a cubic equation of the form `y=ax^3+bx^2+cx+d` into the form `y=a(x-h)^3+k` .

(1) Lets expand the vertex form:

`y=a(x-h)^3+k`

`=a[x^3-3x^2h+3xh^2-h^3]+k`

`=ax^3-3ahx^2+3ah^2x-ah^3+k`

`=ax^3+(-3ah)x^2+(3ah^2)x+(k-ah^3)`

(2) Equate the corresponding coefficients with the equation in standard form, thus:

`a=a`

`b=-3ah`

`c=3ah^2`

`d=k-ah^3`

(3)** Then the required conversions are given by:**

`a=a`

`h=b/(-3a)`

`k=d+ah^3`

Example: Given `y=3x^3-18x^2+36x-20` we find:

`a=3`

`h=(-18)/(-3(3))=2`

`k=-20+3(2)^3=-20+24=4`

So the vertex form is `y=3(x-2)^3+4` .