Question Number 114253 by Dat_Das last updated on 18/Sep/20

For a cubic function in the form:  f(x) = ax^3 +bx^2 +cx+d  What must be true of a, b, c, and d in  order for the function to be able to be  converted to the form:  f(x) = a(x−h)^3 +k

Answered by 1549442205PVT last updated on 18/Sep/20

We need must find a,b,c,d such that   ax^3 +bx^2 +cx+d=a(x−h)^3 +k  ⇔ ax^3 +bx^2 +cx+d≡ax^3 −3ahx^2 +3ah^2 x−ah^3 +k  ⇔ { ((b=−3ah)),((c=3ah^2 )),((d=−ah^3 +k)) :}  where a is arbitrary;h,k given

Commented byDat_Das last updated on 18/Sep/20

This is true, however you may relate  a, b, c directly. Take  b=−3ah  (b/(−3a))=h  (b^2 /(9a^2 ))=h^2   (b^2 /(3a))=3ah^2   Substituting c for 3ah^2  we get  (b^2 /(3a))=c  Therefore for any function  ax^3 +bx^2 +cx+d that may have the form  a(x−h)^3 +k, then  c=(b^2 /(3a))

Commented by1549442205PVT last updated on 18/Sep/20

ThankYou.But d is alone!it isn′t  arbitrary!so c=(b^2 /(3a)) isn′t enough!

Commented byDat_Das last updated on 18/Sep/20

I thought about this. After checking that  c=(b^2 /(3a)) is true and determining h, you set  k=d+ah^(3 ) . This accounts for the only  specification of d and makes it independent  of a, b, c.

Commented byRasheed.Sindhi last updated on 18/Sep/20

Couldn′t we determine relation  between a,c,d ; b,c,d or a,b,d by  excluding one of a,b,c ?