How convert a cubic equation in standard form ax^3+bx^2+cx+d to vertex form a(x-h)^3+k I need to know how to algebraically convert from standard form to vertex form not just say look of graph I am provided with with standard form and need to convert it to vertex form using algebra Thank You so much!
- print Print
- list Cite
Expert Answers
Eric Bizzell
| Certified Educator
briefcaseTeacher (K-12)
calendarEducator since 2011
write3,149 answers
starTop subjects are Math, Science, and Business
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` .
Related Questions
- Find the cubic polynomial f(x)=ax^3+bx^2+cx+d that has horizontal tangents at the points (-1,-6)...
- 1 Educator Answer
- Show that, for any cubic function of the form y= ax^3+bx^2+cx+d there is a single point of...
- 1 Educator Answer
- Find the cubic polynomial f(x) = ax^3 + bx^2 + cx + d that has horizontal tangents at the points...
- 1 Educator Answer
- a quadratic function has zeros at 1 and -3 and passes through the point (2,10). Write the...
- 1 Educator Answer
- Calculate f(-7) if f(7)=7 f(x)= ax^7 + bx^3 + cx - 7
- 1 Educator Answer