In Problem 46 through 55, Find all the cubics through the given points. You may use the results from Exercises 44 and 45 throughout. If there is a unique cubic, make a rough sketch of it. If there are infinitely many cubics, sketch two of them.
Thus, the cubic that passes through the nine given points is of the form .
Each point defines an equation in the 10 variables given by:
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There are nine points.
The system of nine equations is written as follows:
Plug in the nine points to derive the A matrix.
Now, use gauss-Jordan elimination to solve the system . Note that the A matrix is identical to the A matrix from Exercise 52, with the addition of one row. Thus, the first eight rows is replaced with row echelon form in Exercise 52.
The solution of the equation which satisfies:
While are free variables. Recall that the cubic equation is as follows:
Therefore, the cubic that passes through the nine given points is of the form
As the first example, substitute. The cubic is
As the first example, substitute. The cubic is.
Now, for a pointon the cubic curve is either. This set is graphed as follows:
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