Flight of Bird, Cubic Function

[CR7]

Homework Statement

A bird dove under water and re-emerged with a fish. Following is the table which shows the bird's estimated flight:

Time(s),Height(m)
0, 7
2, 10
4, 5
6, 0
7, 0
8, 3

Find a cubic equation to model the data.

2. The attempt at a solution

y=a(x-6)(x-7)(x-k) a,k are both unknown variables

5=a(4-6)(4-7)(4-k) Substituted any 2 points from the table above

5=a(-2)(-3)(4-k)

5=6a(4-k)

Not sure how to solve because there are 2 unknowns. Am I on the right track?

 

[bossman27]

You really only substituted 1 point, defined by two values (t, h).

Try substituting one more different point into the original, then you'll have two equations with two unknowns.

 

[LCKurtz]

@cr7: What level of math are you taking? Are you given that the points come from a cubic so a cubic through any 4 points will automatically fit the others? Or have you studied least squares fitting? Of course, if the points are exactly from a cubic, least squares fit will give it too.